Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol106.Issue2.1999:
Journal of Computational and Applied Mathematics 106 (1999) 271–297
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Some functions that generalize the Krall–Laguerre polynomials
F. Alberto Grunbauma; ∗; 1 , Luc Haineb; 2 , Emil Horozovc; 3
a
b
Department of Mathematics, University of California, Berkeley, CA 94720, USA
Department of Mathematics, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
c
Department of Mathematics and Informatics, So a University, So a 1126, Bulgaria
Received 29 December 1998
Abstract
Let L( ) be the (semi-in nite) tridiagonal matrix associated with the three-term recursion relation satis ed by the
1
Laguerre polynomials, with weight function ( +1)
z e−z ; ¿ − 1, on the interval [0; ∞[. We show that, when is a
positive integer, by performing at most successive Darboux
Pk transformations from L( ), we obtain orthogonal polynomials
1
on [0; ∞[ with ‘weight distribution’ ( −k+1)
z −k e−z + j=1 sj (k−j) (z), with 16k6 . We prove that, as a consequence
of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a ( nite order) di erential
operator. Our construction calls for a natural bi-in nite extension of these results with polynomials replaced by functions,
c 1999 Elsevier Science B.V. All rights reserved.
of which the semi-in nite case is a limiting situation.
1. Introduction
The bispectral problem, as originally posed and solved by Duistermaat and Grunbaum [8], consists
2
in nding all Schrodinger operators L = ddx2 + V (x), for which some families of eigenfunctions f(x; z ),
satisfying Lf = zf, are also eigenfunctions of a di erential operator of arbitrary (but xed) order
in the spectral variable z; Bf = (x)f. The complete solution of this problem revealed its intimate
connection with the theory of Darboux transformations and integrable systems. In a nutshell, all
solutions of the problem can be obtained by means of repeated application of the Darboux process
to some of its basic solutions. By a ‘basic’ solution, we mean a solution for which the bispectral
operator B is of lowest possible order, in this case of order 2.
∗
Corresponding author.
Supported in part by NSF Grant # DMS94-00097 and by AFOSR under Contract FDF49620-96-1-0127.
2
A Research Associate of the Belgian National Fund for Scienti c Research.
3
Supported in part by Grant MM-523 of Bulgarian Ministry of Education.
1
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 6 9 - 2
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
In [13] some of us formulated a discrete–continuous version of the original problem, where the
Schrodinger operator is replaced by a doubly in nite tridiagonal matrix
..
.
L=
..
.
a−2
..
.
b−1
a−1
1
b0
a0
1
b1
a1
1
b2
a2
1
b3
..
.
1
..
.
..
.
;
(1.1)
with the convention that b1 is the (0; 0)th entry of L. More precisely, the problem consists in
determining all bi-in nite tridiagonal matrices L, such that at least one family of eigenfunctions
fn (z ); n ∈ Z, given by
an fn−1 (z ) + bn+1 fn (z ) + fn+1 (z ) = zfn (z );
(1.2)
is also a family of eigenfunctions of a di erential operator of order m (with coecients independent
of n ∈ Z):
m
X
d
d i fn (z )
B z;
= n fn (z ):
fn (z ) ≡
hi (z )
dz
d zi
i=0
(1.3)
The ‘basic’ solutions correspond again to the case m = 2. They are obtained by shifting n to
n + in the three-term recursion relation satis ed by either the Hermite, the Laguerre, the Jacobi
or the (lesser known) Bessel polynomials, letting n run over all integers, see [13,16]. Observe
that the ‘associated polynomials’, see [1,2], satisfy the corresponding three-term recursion relation.
However, as soon as 6= 0, the family of common solutions to Eqs. (1.2) and (1.3) is not given by
the associated polynomials, but rather by functions which can be speci ed in terms of an arbitrary
solution of Gauss’ hypergeometric equation or some of its con
uences, see [13]. It is only when =0,
that one can put f−1 (z ) = 0 and f0 (z ) = 1 and replace the matrix L in Eq. (1.1) by the semi-in nite
matrix obtained by chopping all the columns to the left of b1 and all the rows above b1 . In this case
the fn ’s become the Hermite, the Laguerre, the Jacobi and the Bessel polynomials. This gives back
the classical result of Bochner [6], characterizing the classical orthogonal polynomials as the only
families of orthogonal polynomials, which are eigenfunctions of a second order di erential operator.
The more general problem of determining all orthogonal polynomials which are eigenfunctions of
a di erential operator of arbitrary order was formulated by Krall [24], back in 1938. He found that
the operator had to be of even order and, in [25], he gave the complete solution for m = 4. The new
orthogonal polynomials that he discovered are now coined under the name of ‘Krall polynomials’.
The Krall polynomials are strongly related to some instances of the classical orthogonal polynomials and, in [12], Grunbaum and Haine showed that they can be obtained from some instances
of the Laguerre and the Jacobi polynomials by one or two applications of the (matrix) Darboux
process at the end points of their interval of orthogonality. At this point, the reader will have noticed the absence of the Hermite and the Bessel polynomials as ‘building blocks’ for corresponding
Krall polynomials. The ‘mystery’ was partially elucidated in [15], by showing how to produce out
of the Hermite and any instance of the Bessel polynomials, a pentadiagonal matrix with a family
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
273
of polynomial eigenfunctions that satis es a xed di erential equation of order 4. The same construction can be performed out of any instance of the Jacobi and the Laguerre polynomials [14];
the Krall polynomials are distinguished as the special cases when the above pentadiagonal matrix
is the square of a tridiagonal matrix. This never happens in the case of the Hermite and the Bessel
polynomials.
In this paper we show that repeated application of the Darboux process starting from the semiin nite matrix L( ), associated with the three-term recursion relation satis ed by the Laguerre polynomials, for positive integer values of its parameter , leads to orthogonal polynomials which are
eigenfunctions of a ( nite order) di erential operator. Actually, we will see that it is natural to
enlarge the problem to the two-parameters bi-in nite extension L( ; ) of the matrix L( ) mentioned
above, which reduces to it when = 0. When =
6 0, we obtain in this way functions which provide
higher order instances of solutions of the discrete-continuous version of the bispectral problem, and
which have never appeared before, even in the case of order 4.
One of the key new ideas, that has been introduced in the area of the bispectral problem in
the last few years, is due to Wilson [31,32], who raised the original question at the level of a
commutative algebra of di erential operators, that is at the level of the ‘common eigenfunctions’ of
the algebra. This led him to introduce the seminal idea of the bispectral involution, which amounts
to interchanging the role of the ‘space’ and the ‘spectral’ variables in these eigenfunctions. In
this way, he obtained a complete description of all (maximal) rank one commutative algebras of
bispectral di erential operators. Recently, his ideas were further developed in the works of Bakalov
et al. [3–5] and Kasman and Rothstein [17], aiming at obtaining further examples of higher order
rank commutative algebras of bispectral di erential operators, the rst examples of which appeared
already in the original work of Duistermaat and Grunbaum [8]. Although the methods of [8,31]
appeared to be very di erent, these works allowed to see them as part of a general theory, by
introducing the concept of a bispectral Darboux transformation. For the study of some new and
intriguing examples of bispectral ordinary di erential operators, of a di erent nature than those in
[8], we refer the reader to [4,5,11].
An important message of this paper is to show that the bi-in nite extension of Bochner’s original
result [6], obtained in [13], is the natural context in which Wilson’s ideas can be adapted to the
discrete-continuous version of the bispectral problem. Indeed, only bi-in nite tridiagonal matrices
possess a two-dimensional kernel, from which bispectral Darboux transformations can be performed.
In our context, Wilson’s bispectral involution becomes an anti-isomorphism from an algebra of
matrices to an algebra of di erential operators. This anti-isomorphism is given explicitly by the
three-term recursion relation, the di erentiation formula and the second-order equation which are
satis ed by the two-dimensional space of ‘bispectral functions’, corresponding to the ‘basic’ solutions
of the problem.
Section 2 describes the two-dimensional space of common eigenfunctions to Eqs. (1.2) and (1.3),
when the matrix L in Eq. (1.1) is the bi-in nite Laguerre matrix L( ; ). The use of one Darboux
transformation is presented in detail in Section 3, where we obtain results that had been previously
obtained by other methods when = 0. This will prepare the reader for the more technical Section 4
which is concerned with the iteration of the Darboux process. When =0 and the number of Darboux
transformations is less than +1; ¿ 0, we obtain in this way orthogonal polynomials which extend
Koornwinder’s generalized Laguerre polynomials [20] (see also [18]), with weight distributions
involving not only the delta function, but also its derivatives. If we start the Darboux process at a
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
positive integer value of , these orthogonal polynomials are eigenfunctions of a di erential operator
and, to the best of our knowledge, this result is new. When 6= 0, there is a family of functions
which solve simultaneously Eqs. (1.2) and (1.3) and become polynomials only when = 0. We
propose to call these functions the Krall–Laguerre functions. Finally, in Section 5, we illustrate the
theory by considering the simplest example which in the limit → 0 leads to orthogonal polynomials
with weight distribution involving not only the delta-function but also its rst derivative.
2. The bi-in nite Laguerre matrix and its associated bispectral triple
The bi-in nite Laguerre matrix L( ; ) (in short, L) is a tridiagonal matrix as in Eq. (1.1) which
is obtained by shifting n to n + in the coecients of the standard recursion relation which de nes
the (monic) Laguerre polynomials, with the understanding that n runs over all integers:
an = (n + )(n + + );
bn = 2(n + ) + − 1:
(2.1)
In the sequel, we shall denote by L( ) the semi-in nite matrix which is obtained by chopping all the
columns to the left of b1 and all the rows above b1 in L( ; 0), and de nes the three-term recursion
relation satis ed by the usual (monic) Laguerre polynomials.
It is shown in [13] (see also [16]) that there is a two-dimensional space of functions {fn (z )}n ∈ Z
satisfying the following three properties:
(i) {fn (z )} satisfy the three-term recursion relation
zf = L( ; )f;
(2.2)
with f ≡ (: : : ; f−1 ; f0 ; f1 ; : : :)T ;
(ii) {fn (z )} satisfy a di erentiation formula
A z;
d
f = Mf;
dz
(2.3)
with A a rst-order di erential operator and M a (bi-in nite) tridiagonal matrix;
(iii) {fn (z )} are eigenfunctions of a second-order di erential operator
B z;
d
f = f;
dz
(2.4)
with the diagonal matrix of eigenvalues of B, = diag(: : : ; −1 ; 0 ; 1 ; : : :).
The data listed in (i) – (iii) can be described as follows. Pick w(z ) an arbitrary solution of the
equation
zw′′ (z ) + ( + 1 − z )w′ (z ) + w(z ) = 0;
(2.5)
which is obtained from Gauss’ hypergeometric equation by con
uence. De ne f0 (z ) and f1 (z ) by
f0 (z ) = w(z );
(2.6)
f1 (z ) = (z − − 1 − )w(z ) − zw′ (z ):
(2.7)
and
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
275
Then, the family of functions {fn (z )}n ∈ Z de ned by the three-term recursion relation (2.2) with
f0 (z ) and f1 (z ) as in Eqs. (2.6) and (2.7), satis es automatically Eqs. (2.3) and (2.4) with
d
A=z ;
dz
(2.8)
Mn; n+1 = 0; Mn; n = n + ; Mn; n−1 = (n + )(n + + );
d2
d
+ (z − − 1) ;
2
dz
dz
n = n + :
B = −z
(2.9)
One can always pick [9, 6.3(2)]
w(z ) =1 F1 (−; + 1; z );
(2.10)
to be a solution of Eq. (2.5), with 1 F1 (a; c; z ) denoting the hypergeometric series
1 F1 (a; c; z ) =
∞
X
(a)n
n=0
n!(c)n
zn :
(2.11)
Here, as well as in the rest of the paper, (x)n , n ∈ Z, is the shifted factorial (Pochammer notation):
(x + n)
(x)0 = 1; (x)n =
= (x + n − 1)(x)n−1 :
(2.12)
(x )
It follows from [9, 6.4(2),6.4(9)] that the solution of the three-term recursion relation (2.2) with
initial conditions given by f0 (z ) and f1 (z ) as in Eqs. (2.6) and (2.7), and w(z ) as in Eq. (2.10), is
given by
fn (z ) = (−1)n ( + 1 + )n 1 F1 (−n − ; + 1; z ):
(2.13)
Notice that, when = 0, the functions fn (z ), n¿0, reduce precisely to the Laguerre polynomials,
normalized to be monic, see [9, 10.12(14)]. For this reason, we shall call the functions fn (z )
de ned in Eq. (2.13) the Laguerre functions. They are eigenfunctions of the second order di erential
operator B de ned in Eq. (2.9), which generalizes the standard second order di erential equation
satis ed by the Laguerre polynomials, and they will play a basic role in what follows. As emphasized
in the introduction, we notice that, although the three-term recursion relation satis ed by the functions
fn (z ) is the same as the one satis ed by the associated Laguerre polynomials studied in [2], when
6= 0, the fn (z ) are not polynomials. The ‘associated polynomials’ satisfy Eq. (2.2) but, for =
6 0,
do not satisfy Eq. (2.4).
We shall denote by
B = hL; M; i;
(2.14)
the subalgebra of the algebra of nite band bi-in nite matrices generated by the matrices L, M ,
appearing on the right-hand side of Eqs. (2.2) – (2.4). Similarly
B′ = hz; A; Bi;
(2.15)
will denote the subalgebra of the algebra of di erential operators generated by z , A, B. Formulas
(2.2) – (2.4) serve to de ne an anti-isomorphism
b : B → B′
(2.16)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
between these two algebras, i.e. it is given on the generators by
b(L) = z;
b(M ) = A
and
b() = B:
(2.17)
More precisely, any monomial Li M j k in B, i; j; k¿0, acting on the original space of bispectral
functions {fn }n ∈ Z , gives
Li M j k f = Bk Aj z i f;
i.e.
b(Li M j k ) = b(k )b(M j )b(Li ) = Bk Aj z i :
(2.17′ )
The triple (B; B′ ; b) provides an instance of the notion of a bispectral triple which was introduced
in [4]. To explain the terminology, we need to introduce the commutative subalgebras K ⊂ B and
K′ ⊂ B′ generated respectively by and z . We shall refer to these subalgebras as the ‘algebras
of functions’. Their images by b and b−1 will be denoted by A′ and A respectively and provide
obvious bispectral operators. In order to de ne the notion of Darboux transformation in this abstract
setting, we shall denote with a bar the elds of quotients of K, K′ , A and A′ . Obviously, b
′
′
extends to isomorphisms K → A and A → K . We now reproduce from [15] the main tool that
we need to produce out of L( ; ) new non-trivial bispectral operators by means of the Darboux
transformation, see also [4].
Theorem 1. Let L ∈ A be a constant coecients polynomial in L; which factorizes ‘rationally’
as
L = QP;
(2.18)
in such a way that
Q = SV −1 ;
P = −1 R;
(2.18′ )
with R; S ∈ B and V ∈ K. Then the Darboux transform of L given by
˜ = PQ;
L
(2.19)
is a bispectral operator. More precisely; de ning ≡ b(L) ∈ K′ and f˜ ≡ Pf; with f satisfying
Eqs. (2:2)–(2:4) above; we have
˜
˜ f˜ = f;
L
(2.20)
˜
B˜ f˜ = V f;
(2.21)
B˜ = b(R)b(S )−1 :
(2.22)
with
Proof. Eq. (2.20) follows immediately from the de nitions. Let
Pb ≡ b(R)
and
Qb ≡ b(S )−1 :
(2.23)
′
Clearly, using the anti-isomorphism introduced in Eqs. (2.16), (2.17), (2:17 ),
f˜ = Pf = −1 Pb f:
(2.24)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
277
Since B has no zero divisors, Eqs. (2.18), (2.18′ ) imply
V = RL−1 S; L−1 ∈ A:
Applying the anti-isomorphism b, we obtain
b(V ) = b(S )−1 b(R):
(2.25)
From Eqs. (2.23) – (2.25), we have
˜
Vf = Qb Pb f = Qb f;
and thus, using Eq. (2.18) and this last relation, we deduce that
˜
f = −1 Qf˜ = V −1 Qb f:
(2.26)
This equation combined with Eqs. (2.23) and (2.24) gives Eq. (2.21), which completes the proof
of the theorem.
Remark. Observe that B˜ in Eq. (2:22) is a Darboux transformation of b(V ) in Eq. (2:25). Since
V ∈ K; V is a polynomial in ; and thus b(V ) is a polynomial in B. This shows that the
new bispectral operator B˜ is in fact obtained as a Darboux transform of a constant coecients
polynomial in the original (second order) bispectral operator B.
Notice that Theorem 1 leaves open the question of when a ‘rational’ factorization of the form
(2.18), (2.18′ ) can be performed. Our aim in the next two sections is to show that, for any positive
integer , the successive powers L( ; )k , k = 1; 2; : : : ; , of the doubly in nite Laguerre matrix admit
˜ in Eq. (2.19) is again the
such a factorization with the additional property that the new operator L
−1
˜
k th-power of a tridiagonal matrix. The functions fn = (Pf)n = Pb fn , obtained from the Laguerre
functions fn in Eq. (2.13), are thus solutions of the discrete-continuous version of the bispectral
problem (1.2), (1.3), with a bispectral operator B of order m ¿ 2. We shall refer to them as the
Krall–Laguerre functions. When = 0, these functions become polynomials which generalize the
Krall–Laguerre polynomials. For a precise de nition, see Theorem 3 in Section 4.
3. One step of the Darboux process
In this section we rst perform a standard Darboux transformation on the doubly in nite Laguerre
matrix L( ; ), when ¿ 0 and is arbitrary. In the limit = 0, we obtain in this way orthogonal
1
z −1 e−z +
(z ), where
is the free parameter
polynomials on the interval [0; ∞[ for the measure ( )
of the Darboux transformation. These polynomials (as well as an extension to the Jacobi case) were
found by Koornwinder [20]. In particular, he proved that, in general, they satisfy a second-order
di erential equation with coecients depending on n. In the special case when is a positive
integer, we show in Section 3.2 that the Darboux factorization of L( ; ) can be recast in the form
(2.18), (2.18’) called for in Theorem 1, and as a consequence, the Darboux transform of L( ; )
is again bispectral, in the sense of Eqs. (1.2) and (1.3). When = 0 and = 1, we get back the
classical Krall–Laguerre polynomials, discovered by Krall [25]. When ¿ 1, this result has been
obtained (using other methods) by Littlejohn [29] when = 2, by Krall and Littlejohn [23] when
= 3, and by Koekoek and Koekoek [18] for an arbitrary positive integer . The case 6= 0 appears
here for the rst time and leads to ‘Krall–Laguerre functions’.
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
3.1. Factorizing the bi-in nite Laguerre matrix
We remind the reader that the standard Darboux transformation (referred to in the sequel as
elementary Darboux transformation) of a bi-in nite tridiagonal matrix L [30] starts by factorizing L
as
L = QP;
(3.1)
with the two factors Q and P denoting, respectively, upper and lower triangular matrices acting on
a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T as follows:
(Qh)n = x n+1 hn + hn+1 ;
(3.2)
(Ph)n = yn hn−1 + hn :
(3.3)
The most general factorization of L, in the form (3.1), is obtained by picking an arbitrary element
f ∈ ker L, so that the matrices P and Q are given by
fn
hn−1 ;
(3.4)
(Ph)n = hn −
fn−1
fn−1
hn + hn+1 :
(3.5)
fn
Since the kernel of L is two-dimensional and only the ratios of the fn ’s are involved, this factorization depends (projectively) on one free parameter. The Darboux transformation L˜ of L is obtained
by exchanging the order of the factors in Eq. (3.1)
L˜ = PQ:
(3.6)
(Qh)n = −an
Explicitly, the entries a˜n and b˜n of the new tridiagonal matrix L˜ are given by
fn−2 fn
a˜n = an−1 2 ;
fn−1
(3.7)
fn−1
fn
−
:
(3.8)
b˜n = bn +
fn−1 fn−2
As observed in [12], the above construction still makes sense in the case of semi-in nite matrices,
that is if one chops all the columns to the left of the (0; 0)th entry and all the rows above the (0; 0)th
entry of the matrices L, P and Q. The crucial observation is that, in this limit, the upper–lower
factorization (3.1) of L, still contains a free parameter, which can be picked to be x1 . This would
no longer be the case, if instead we had chosen to perform a lower–upper factorization of L, in
which case all the entries of P and Q are uniquely speci ed.
Thus, in order to perform the (most general) factorization of the form (3.1) for the bi-in nite
Laguerre matrix L( ; ), we need a description of its two-dimensional kernel. Introducing the lower
shift matrix
(T −1 h)n = hn−1 ;
(3.9)
and the matrix C de ned by
(Ch)n = (n + 1 + )hn + hn+1 ;
(3.10)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
279
one checks easily that
L( ; ) = T −1 C (C + I );
(3.11)
with I the identity matrix. From this important observation, since the matrices C and C + I commute,
we deduce immediately the next result.
Lemma 3.1. Let ¿ 0. The kernel of L( ; ) is generated by the vectors = (: : : ; −1 ; 0 ; 1 ; : : :)T
and = (: : : ; −1 ; 0 ; 1 ; : : :)T ; where
n = (−1)n (1 + )n ;
n
(3.12)
= (−1)n (1 + + )n ;
(3.13)
and (x)n denotes the shifted factorial.
Corollary 3.2. The most general factorization of the bi-in nite Laguerre matrix L( ; ) in the
form (3:1)–(3:3) is given by
an−1
n +
n
;
xn =
;
(3.14)
yn = −
n−1 +
n−1
yn−1
with
an arbitrary free parameter; and n ;
n;
an as in Eqs. (3:12); (3:13) and (2:1).
Proof. From Lemma 3.1, we have that
f ∈ ker L( ; ) ⇔ f = +
;
with
an arbitrary free parameter (we allow
= ∞; i.e. f = ), which using Eqs. (3.4) and(3.5)
gives Eq. (3.14) and establishes the corollary.
One checks easily that in the limit → 0; y0 → 0 and x1 → = (1 +
); and thus one obtains a
factorization of the semi-in nite Laguerre matrix, where x1 is equivalent to the free parameter
.
The Darboux transform L˜ (3.6) de nes new orthogonal polynomials qn −1;
given by
qn −1;
= (Pp )n
= pn +
n! +
( + 1)n
p ;
(n − 1)! +
( + 1)n−1 n−1
1
z e−z on [0; ∞[;
where pn (z ) denote the (monic) Laguerre polynomials with weight function ( +1)
when ¿ − 1. Using the standard formulas (see, for instance, [9, 10.12(15),10.12(16)])
1 d −1
d
=
pn−1
pn −1 ;
and pn = 1 −
pn
n dz
dz
we get that
d −1
( )n
qn −1;
= pn −1 +
p ;
n! +
n( + 1)n−1 d z n
which precisely agrees with the formula given in [18,20] for Koornwinder’s generalized Laguerre
1
polynomials (normalized to be monic) with weight function ( )
z −1 e−z +
(z ) on the interval
[0; ∞[; when ¿ 0 and
¿0.
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
3.2. Factorizing in the algebra B; when is a positive integer
In the special case when is a positive integer, it is possible to achieve the form of the factorization (2.18), (2:18′ ); called for in Theorem 1. For this, it will be convenient to use another basis
of the algebra B = hL; M; i introduced in Eq. (2.14), namely
B = hN; T; i;
(3.15)
where N denotes the strictly lower part of the bi-in nite Laguerre matrix L = L( ; ):
(Nh)n = an hn−1 = (n + )(n + + )hn−1 ;
(3.16)
and T is the upper shift matrix
(Th)n = hn+1 :
(3.17)
One checks easily that the change of basis is given by
N = M − ;
T = L − − M − ( + 1)I:
(3.18)
The point of this change of basis is that now the three generators N; T; are lower, upper and
diagonal matrices, respectively, with only one non-zero diagonal. They give the most convenient
way to factorize a tridiagonal matrix. Using these new generators for B; we have
Lemma 3.3. Let be a positive integer. Then; the Darboux factorization (3:14) of L( ; ) can
be recast into the form called for in Theorem 1:
L( ; ) = QP ≡ (SV −1 )(−1 R);
(3.19)
with R; S ∈ B given by
R = ( + I )() + N( + 2I );
(3.20)
S = ( + I )() + T( + I );
(3.21)
= ()( + I );
(3.22)
and
V = ( + I );
where () a polynomial of degree in
( ) = I +
−1
Y
( + jI );
j=0
=
:
(1 + )
(3.23)
Proof. Using that is a positive integer and the de nition of an in Eq. (2.1), the entries yn and x n
(3.14) of the factors Q and P in the Darboux factorization (3.1) become rational functions of n:
(n + )(n + + 1)
(n − 1 + + )(n + −1)
yn =
;
xn =
;
(3.24)
(n + )
(n + )
with (n) a polynomial of degree in n:
(n) = 1 + (n) ;
=
= (1 + ) :
(3.25)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
281
Since
(Ph)n = hn + yn hn−1
1
=
((n + )hn + (n + )(n + + 1)hn−1 );
(n + )
(3.26)
remembering that is the diagonal matrix = diag(n ); n = n + ; with 0 = at the (0; 0)th entry,
we can rewrite P as follows
P = ()−1 [() + T −1 ( + I )( + 2I )]:
(3.27)
The (lower) shift matrix T −1 does not belong to the algebra B; but we observe that, by the de nition
of the an ’s (2.1), the matrix N ∈ B (3.16) can be written as
N = T −1 ( + I )( + (1 + )I )
= ( + I )T −1 ( + I ):
(3.28)
Therefore,
P = ()−1 [() + ( + I )−1 N( + 2I )]
= ()−1 ( + I )−1 [( + I )() + N( + 2I )]:
(3.29)
Similarly,
(Qh)n = x n+1 hn + hn+1
= (n + + )
(n + )
hn + hn+1 ;
(n + + 1)
(3.30)
and thus,
Q = ( + I )()( + I )−1 + T
= [( + I )() + T( + I )]( + I )−1 :
(3.31)
Since the (upper) shift matrix T belongs to B; the factor between the brackets on the right-hand
side of the last equality belongs automatically to B. Combining Eqs. (3.25), (3.29) and (3.31) gives
the result announced in the lemma, which concludes the proof.
Notice that in this case, both and V in Eq. (3.22) belong to the algebra of ‘functions’ K
(i.e. the constant coecients polynomials in ), and thus the product V ∈ K. Since we already
observed that R and S in Eqs. (3.20) and(3.21) belong to the algebra B; using Theorem 1 with
= b(L( ; )) = z; we deduce immediately
Corollary 3.4. Let be a positive integer. The Darboux transform L˜ = PQ of L( ; ); with P; Q
in Eq. (3:19) is again bispectral. Explicitly; de ning
f˜n = (Pf)n ;
with fn as in Eq. (2:13); we have
L˜f˜n = z f˜n ;
B˜ f˜n = (n + + )(n + )(n + + 1)f˜n ;
(3.32)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
with
B˜ = b(R)b(S )z −1 ;
(3.33)
where b : B → B′ is the bispectral anti-isomorphism de ned in Eqs. (2:16); (2:17); (2:17′ ).
To see that these formulas are actually constructive, we compute from Eqs. (2.17) and (3.18)
b() = B; b(N ) = A − B;
b(T ) = z − B − A − ( + 1);
(3.34)
and then, using the fact that b in Eq. (2.16) is an anti-isomorphism, we get from Eqs. (3.20), (3.21)
that
b(R) = (B)(B + ) + (B + 2)(A − B);
(3.35)
b(S ) = (B)(B + ) + (B + 1)(z − B − A − ( + 1)):
Recalling the remark following Theorem 1, B˜ is a Darboux transform of b(V ). Since (V )n =
(n + + )(n + )(n + + 1) is a polynomial of degree 2 + 1 in n + ; both b(V ) and B˜ in Eq.
(3.33) are operators of order 2(2 + 1). This order is larger than the order of the operator produced
in [18]. The issue of getting the lowest possible order will be discussed in Section 5.
4. Iterating the Darboux process
In this section we rst describe the result of k iterations of the Darboux process starting from
the bi-in nite Laguerre matrix L( ; ); ¿ 0; k ¡ + 1. We compute the moment functional for the
resulting orthogonal polynomial sequence, when = 0. Then, in Section 4.2, when is a positive
integer and the number of elementary Darboux transformations is less than or equal to ; we show
that the process leads to functions which solve simultaneously Eqs. (1.2) and (1.3). In the limit =0;
these functions become orthogonal polynomials which are eigenfunctions of a di erential operator
and generalize the Krall–Laguerre polynomials; for this reason, we call them the Krall–Laguerre
functions.
4.1. Extending Koornwinder’s generalized Laguerre polynomials
In order to compute the e ect of k successive elementary Darboux transformations starting from
L0 ≡ L( ; )
L0 = Q0 P0 y L1 = P0 Q0 = Q1 P1 y : : :
y Lk−1 = Pk−2 Qk−2 = Qk−1 Pk−1 y Lk = Pk−1 Qk−1 ;
(4.1)
we need the following lemma, which follows easily from the representation (3.11) of L( ; ) and
generalizes Lemma 3.1:
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
j)
Lemma 4.1. For j¿0; the vectors ( j) =(: : : ; (−1
; (0j) ; (1j) ; : : :)T and
with entries
(nj) = (−1)n+j
( j)
n
= (−1)n+j
( j)
=(: : : ;
( j)
−1 ;
( j)
0 ;
283
( j)
T
1 ; : : :) ;
(1 + )n+j
;
j !(1 − )j
(4.2)
(1 + + )n+j
;
j !(1 + )j
(4.3)
satisfy
L( ; ) ( j) = ( j−1)
and
L( ; )
with the natural interpretation (−1) =
( j)
=
(−1)
( j−1)
;
(4.4)
= 0.
Let 0 ; : : : ; k−1 ;
0 ; : : : ;
k−1 denote free parameters and, for 06j6k − 1, de ne
f
( j)
=
j
X
( l ( j−l) +
l
( j−l)
):
(4.5)
l=0
Since, from Lemma 4.1,
L0 f( j) = f( j−1) ;
(4.6)
one shows inductively that, for 06j6k − 1,
ker Pj = (one-dimensional) span of Pj−1 Pj−2 : : : P0 f( j) :
(4.7)
When j = 0, this formula is interpreted as ker P0 = f(0) , reproducing the result given in Lemma 3.1.
Thus, the product matrix P
P = Pk−1 Pk−2 : : : P0 ;
(4.8)
has a kernel generated by f(0) ; f(1) ; : : : ; f(k−1) , and therefore it can be written as the ratio of two
Casorati determinants (using the action on a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T ):
(0)
(k−1)
fn−k
: : : fn−k
hn−k
..
..
.
..
.
.
(k−1)
f(0)
:
:
:
f
h
n
n
(Ph)n = n (0)
(k−1)
: : : fn−k
fn−k
.
..
.
.
.
(0)
(k−1)
fn−1
: : : fn−1
:
(4.9)
Another way to put this is to say that the sequence of elementary Darboux transformations (4.1)
can be performed in one shot by factorizing
L ≡ Lk0 = QP;
(4.10)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
with P as in Eq. (4.9) and Q a uniquely determined upper triangular matrix with k + 1 diagonals,
with the entries on the top diagonal normalized to be one. Then, the Darboux transform is
˜ = PQ = (Lk )k ;
L
(4.11)
with Lk the tridiagonal matrix obtained at the end of the chain of the elementary Darboux transformations (4.1). Indeed, since
PL0 = Lk P;
we deduce that
˜
(Lk )k = P (L0 )k P −1 = PQPP −1 = L:
Remark 4.2. From Eq: (4:9) we see that two sequences of parameters ( l ;
l ) and ( l′ ;
′l ); 06l6k−
1; lead to the same factorization (4.10) as long as the corresponding functions f( j) in Eq: (4:5);
06j6k − 1; generate the same
ag of subspaces hf(0) i ⊂hf(0) ; f(1) i ⊂ · · · ⊂hf(0) ; f(1) ; : : : ; f(k−1) i.
In the basis given by ( j) and ( j) in Eqs: (4:2); (4:3); this
ag is represented by the k × 2k matrix
0
0
:::
1
( ;
) ≡ ...
0
0
..
.
..
.
k−1
1
:::
0
..
.
..
.
1
and can be normalized as
0
1
..
.
0
0
k−1
0
:::
0
0
..
.
..
.
1
:::
..
0
..
.
.
1
;
0
(4.12)
0
(I;
) or ( ; I ) ≡ (
−1 ; I );
(4.13)
depending on whether 0 6= 0 or
0 6= 0; with I the identity matrix. Thus; there are in fact only
k free parameters
0 ; : : : ;
k−1 involved in the factorization (4.10); which can be thought of as the
new free parameters involved in the successive elementary Darboux transformations (4.1).
Our aim now is to compute explicitly the entries of the new tridiagonal matrix Lk in Eq. (4.11).
For this, we need to introduce a few notations that will be repeatedly used in this section.
We can write fn( j) in Eq. (4.5) as
( j)
fn( j) = (1 + )n f˜n ;
(4.14)
with
( j)
f˜n
j
X
(−1) j−l
n
= (−1)
l=0
"
#
(n + + 1)j−l
(1 + + )n+j−l
l
:
+
l
( j − l)!
(1 − )j−l
(1 + )n (1 + )j−l
(4.15)
Now we come to some expressions which will play a crucial role. Section 5 illustrates their usefulness in the simplest case = k = 2. We introduce the determinants
( j)
(k) (n) = det(f˜n−k+i )06i; j6k−1 ;
(4.16)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
˜(0)
fn−k
..
.
(0)
˜
fn−i−1
i (n) = (0)
f˜
n−i+1
.
..
(0)
f˜n
(1)
f˜n−k
..
.
(1)
f˜n−i−1
(1)
f˜n−i+1
..
.
(1)
f˜n
:::
:::
:::
:::
for 16i6k − 1, and put
285
(k−1)
f˜n−k
..
.
(k−1)
f˜n−i−1
;
(k−1)
f˜n−i+1
..
.
(4.17)
(k−1)
f˜n
k (n) = (k) (n + 1):
(4.18)
The following lemma extends Eqs. (3.7) and (3.8) to an arbitrary number of (elementary) Darboux
transformations:
Lemma 4.3. The entries an(k) and bn(k) of the tridiagonal matrix Lk ; obtained from L0 = L( ; ) after
k elementary Darboux transformations; are given by
(k) (n − 1) (k) (n + 1)
;
[ (k) (n)]2
(4.19)
1 (n − 1)
1 ( n )
− (n + − 1) (k)
;
(k)
(n)
(n − 1)
(4.20)
an(k) = (n + )(n + + − k )
bn(k) = bn + (n + )
with bn as in Eq: (2:1).
The following sublemma (whose proof we omit) will be needed to establish Lemma 4.3, as well
as Lemma 4.5 later in this section. Both of these lemmas, whose detailed proofs are given, are used
to establish Theorem 2 and Theorem 3 in this section. They produce very concrete formulas which
will be illustrated in Section 5.
Lemma 4.4. Let V and W be vector spaces of respective codimension 2 and 1 in a vector space E
of dimension k + 2. The (incidence) relation V ⊂ W amounts to the system of quadratic equations
satis ed by the Plucker coordinates p1::: i:::
ˆ jˆ ::: k+2 ; 16i ¡ j6k + 2; (resp. 1::: i:::
ˆ k+2 ; 16i6k + 2) of
V (resp. W ):
1::: i:::
ˆ k+2 p1::: jˆ ::: lˆ::: k+2 − 1 ::: jˆ ::: k+2 p1::: iˆ::: lˆ::: k+2
+1::: lˆ::: k+2 p1::: iˆ::: jˆ::: k+2 = 0;
for all 16i ¡ j ¡ l6k + 2:
(4.21)
Proof. See [10, p. 134, Lemma 2], where the more general case of the incidence relations satis ed
by planes V , W of codimension p and q, p¿q, in a vector space E of dimension n, is discussed.
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
Proof of Lemma 4.3. The proof is done by induction on k . For k = 1, we have that
a(1)
n = an−1
(0)
fn−2
fn(0)
;
(0) 2
[fn−1
]
using Eq: (3:7);
(0)
= an−1
(0)
(1 + )n−2 (1 + )n f˜n−2 f˜n
;
(0)
(1 + )2n−1 [f˜n−1 ]2
by Eq : (4:14);
(1) (n − 1) (1) (n + 1)
;
[ (1) (n)]2
using Eqs: (2:1) and (4:16) with k = 1:
= (n + ) (n + + − 1)
Assume now that formula (4.19) holds for k , we establish it for k + 1. Indeed, since by Eq. (4.7),
the kernel of Lk is spanned by
f = Pk−1 Pk−2 : : : P0 f(k) ;
(4.22)
using Eq. (3.7), we get that
(k) fn−2 fn
an(k+1) = an−1
:
2
fn−1
But, from Eqs. (4.8), (4.9), (4.14), (4.16) and (4.22), we see that
(1 + )n (k+1) (n + 1)
;
(k) (n)
and, therefore, by induction hypothesis,
fn =
(4.23)
(n + )[ (k) (n − 1)]2 (k+1) (n − 1) (k+1) (n + 1)
;
(n − 1 + ) (k) (n − 2) (k) (n)[ (k+1) (n)]2
(k+1) (n − 1) (k+1) (n + 1)
= (n + )(n + + − k − 1)
:
[ (k+1) (n)]2
(k)
an(k+1) = an−1
Similarly, to establish Eq. (4.20), we rst observe that, from Eqs. (3.8) and (4.14), we have that
b(1)
n
(0)
(0)
f˜n−1
f˜n
= bn + (n + ) (0) − (n + − 1) (0) ;
f˜n−1
f˜n−2
with bn as in Eq. (2.1), which using Eqs. (4.16) and (4.18), with k = 1, agrees with Eq. (4.20).
Assume now that formula Eq. (4.20) holds for k , we establish it for k + 1. Indeed, by Eq. (3.8),
we have that
fn−1
fn
−
;
bn(k+1) = bn(k) +
fn−1 fn−2
with fn as in Eq. (4.23). By induction hypothesis, this gives that
(k) (n − 1) (k+1) (n + 1) + 1 (n) (k+1) (n)
(k) (n) (k+1) (n)
(k) (n − 2) (k+1) (n) + 1 (n − 1) (k+1) (n − 1)
−(n + − 1)
:
(k) (n − 1) (k+1) (n − 1)
bn(k+1) = bn + (n + )
(4.24)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
287
Consider now the (k + 2) × (k + 1) matrix
( j−1)
16i6k+2 ;
(f˜n−k−2+i ) 16j6k+1
and the subspaces V (resp. W ) of Rk+2 generated by the rst k columns (resp. all the columns)
of this matrix. Applying Lemma 4.4 to this situation, one checks easily that Eq. (4.21) with i = 1,
j = k + 1 and l = k + 2, amounts to
(k) (n − 1) (k+1) (n + 1) + 1 (n) (k+1) (n) = (k) (n)1(k+1) (n);
with 1(k+1) (n) the (k + 1) × (k + 1) determinant obtained by replacing k by k + 1 into Eq. (4.17).
Thus, Eq. (4.24) coincides with Eq. (4.20), with k replaced by k + 1. This establishes Lemma 4.3.
Remember that, as explained in Section 3, in the case of semi-in nite matrices, the (0; 0)th entry
x1 of Q (3.2) is equivalent to the free parameter of the elementary Darboux transformation. Thus
our construction still makes sense in the limit = 0 and the k free parameters can be thought of to
be the entries x1( j) , 06j6k − 1, of the matrices Qj in Eq. (4.1). The matrix Lk which is obtained at
the end of the chain de nes a new sequence of polynomials q(k) = (q0(k) (z ) = 1; q1(k) (z ); q2(k) (z ); : : :)T
satisfying zq(k) = Lk q(k) . By Favard’s theorem (see, for instance, [7]), as long as the entries an(k) of
Lk are non-zero, there exists a unique (up to a multiplicative constant) moment functional M(k) for
which the sequence {qn(k) (z )}∞
n=0 is an orthogonal polynomial sequence, that is
M(k) [qn(k) qm(k) ] = 0;
for m 6= n
and
M(k) [(qn(k) )2 ] 6= 0:
We remind the reader that the moment functional M corresponding to a sequence of complex
numbers {n }∞
n=0 is a complex valued function de ned on the vector space of all polynomials by
M(z n ) = n ;
n = 0; 1; 2; : : : ;
M(c1 1 (z ) + c2 2 (z )) = c1 M(1 (z )) + c2 M(2 (z ));
for all complex numbers ci and all polynomials i (z ) (i = 1; 2).
The next theorem shows that, iterating the Darboux process starting from the Laguerre polynomials, leads to a natural extension of Koornwinder’s generalized Laguerre polynomials (which
correspond to k = 1).
Theorem 2. The sequence of polynomials {qn(k) (z )}∞
n=0 ; obtained after k successive elementary
Darboux transformations starting from the Laguerre matrix L( ); with k ¡ +1; is an orthogonal
sequence of polynomials with moment functional M(k) given by the weight distribution
k
X
(−1)k−j (k−j)
1
z −k e−z +
rj
(z );
( − k + 1)
(k − j )!
j=1
(4.25)
with
rj =
( − k + 1)k
− ( − k + 1)k−j ;
x1(0) : : : x1( j−1)
(4.26)
where x1(0) ; : : : ; x1(k−1) denote the successive free parameters in the elementary Darboux transformations ((0; 0)th entries of Q0 ; : : : ; Qk−1 ); ( j) (z ) denotes the j th derivative of the delta function and
(a)j is the shifted factorial.
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
Proof. Start with any orthogonal polynomial sequence {pn (z )}∞
n=0 for a moment functional M, with
M[z n ] = n and 0 = 1. One just needs to observe that, after one elementary Darboux transformation,
if the new matrix L˜ in Eq. (3.6) satis es the hypothesis of Favard’s theorem (i.e. a˜n 6= 0, n¿1),
˜
the resulting sequence of polynomials {qn (z )}∞
n=0 de ned by L will be orthogonal for the moment
˜
functional M given by
M˜ [z n ] = x1 n−1 ;
M˜ [1] = 1;
n¿1;
(4.27)
with x1 the free parameter ((0; 0)th entry of Q) of the Darboux transformation. Indeed, since q = Pp,
one also has
zp = Qq ⇔ zpn = x n+1 qn + qn+1 :
Applying M˜ to both sides, we obtain for n = 0 that M˜ [z ] = x1 , and inductively, using the classical
formula expressing the orthogonal polynomials in terms of the moments
1
1
−1
pn (z ) = n−1 ...
n−1
1
1
2
..
.
:::
:::
n
z
:::
:::
n
n+1
..
.
;
2n−1
zn
˜ n
with n−1 = det(i+j )i;n−1
j=0 , one deduces that M[z ] = x1 n−1 .
From (4.27) it follows that, after k elementary Darboux transformations (again assuming that at
each step we can apply Favard’s theorem), the resulting polynomials {qn(k) (z )}∞
n=0 will be orthogonal
for the moment functional M(k) de ned by
M(k) [1] = 1;
M(k) [z j ] = x1(k−1) x1(k−2) : : : x1(k−j) ;
(k)
j
M [z ] =
x1(k−1) x1(k−2)
: : : x1(0) j−k ;
16j6k − 1;
j¿k:
By Lemma 4.3 (formula (4.19), with = 0), we see that as long as k ¡ + 1, an(k) 6= 0 for n¿1,
and thus the argument above applies. Since the moments {sn }∞
n=0 of the weight distribution (4.25)
are given by
sk−j
( − k + 1)k−j + rj
=
; 16j6k − 1;
s0
1 + rk
sj
( − k + 1)k
=
j−k ; j¿k;
s0
1 + rk
we see that this distribution de nes the same moment functional as M(k) , provided that we pick the
rj ’s as in Eq. (4.26), which establishes Theorem 2.
4.2. The bispectral property
We now show that, when is a positive integer and k6 , the factorization (4.10) can be put
into the form (2.18), (2:18′ ) needed to apply Theorem 1. As a consequence, the resulting tridiagonal
matrix Lk obtained after k elementary Darboux transformations will be bispectral. The punch line
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
289
of the argument is to exploit in this case the rational character of the Darboux factorization (4.10).
The denominators from the left-hand side and the right-hand side factors are taken out to produce
part of the ‘function’ V ∈ K. Since the upper-shift matrix T is in the algebra B, what remains
to the left is automatically in B. We expand what remains on the right-hand side in terms of the
lower-shift matrix T −1 , then express the various powers of T −1 in terms of N via Eq. (3.28) and
absorb the denominator in . We now proceed with the details of our program.
Put
p i (n ) ≡
i
Y
(n + j ):
(4.28)
j=1
( j)
Since is a positive integer, we can write f˜n in Eq. (4.15) as
( j)
f˜n = (−1)n g( j) (n + );
(4.29)
with
g( j) (n) =
j
X
(−1) j−l
l=0
where
"
#
l
l
pj−l (n) +
p +j−l (n) ;
( j − l)! (1 − )j−l
(1 + )j−l
l =
l = (1 + ) :
(4.30)
(4.31)
It follows from Eqs. (4.28) – (4.30) that the determinants (k) (n) and i (n), 16i6k , de ned in
Eqs. (4.16) – (4.18), become now polynomials in the variable n + . Thus, we shall write
(k) (n) ≡ 0 (n + );
i (n) ≡ i (n + );
16i6k;
(4.32)
( j)
where the i (n), 06i6k , are the determinants obtained by replacing f˜l by (−1)l g( j) (l) respectively
in Eqs. (4.16) – (4.18).
( j)
Remembering the de nition of f˜n in Eq. (4.14), from each of the rst k columns of both
the numerator and the denominator of Eq. (4.9), we can extract a factor (1 + )n−k . In this way,
expanding the numerator of Eq. (4.9) along the last column, we get
(Ph)n =
k
X
(−1)i
i=0
n
Y
( + j )
j=n−i+1
i (n + )
hn−i ;
0 (n + )
(4.33)
with i (n), 06i6k
, de ned as in Eq. (4.32). Here and in the rest of the paper we make the
Q
convention that nj=n+1 (: : :) = 1.
From the previous formula, it is clear that, in terms of matrices, the operator P can be written as
P = 0 ()−1
k
X
i=0
(−1)i T −i
i
Y
( + jI )i ( + iI ):
(4.34)
j=1
From Eq. (3.28) we deduce easily that
Ni =
i−1
Y
j=0
( + ( − j )I )T −i
i
Y
j=1
( + jI );
(4.35)
290
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
and so, we can rewrite P as
P = −1 R;
(4.36)
with
= 0 ()
k−1
Y
( + ( − j )I );
(4.37)
j=0
R=
k
X
(−1)i
k−1
Y
( + ( − j )I )N i i ( + iI ):
(4.38)
j=i
i=0
It remains to express
Q = Q0 Q1 : : : Qk−1 ;
(4.39)
as Q = SV −1 , with S ∈ B and V a polynomial in . For this we need the following
Lemma 4.5. The operator Q in Eq. (4:39) acting on a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T is given by
(Qh)n =
k
X
n
Y
(−1)i+k
i=0
( + + j )
j=n−k+i+1
i (n + + i)
hn+i ;
0 (n + + i + 1)
(4.40)
with i (n) as in Eq. (4:32).
Proof. The proof is by induction on k . For k = 1, we have that
(Q0 h)n = −an
=−
(0)
fn−1
hn + hn+1 ;
fn(0)
(0)
an f˜n−1
(0)
(n + )f˜n
= −(n + + )
using Eq: (3:5);
hn + hn+1 ;
by Eq: (4:14);
0 (n + )
hn + hn+1 ;
0 (n + + 1)
using Eqs: (2:1); (4:16); (4:32) for k = 1:
Suppose now that the result is true for k elementary Darboux transformations, we deduce it for
k + 1 elementary Darboux transformations. Indeed:
(Qk h)n = −an(k)
fn−1
hn + hn+1 ;
fn
= −(n + + − k )
using Eq: (3:5) with f as in Eq: (4:22);
(k) (n + 1) (k+1) (n)
hn + hn+1 ;
(k) (n) (k+1) (n + 1)
by Eqs: (4:19) and (4:23):
(4.41)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
291
Assuming Eq. (4.40) with Q as in Eq. (4.39), from the formula (4.41) for Qk , we compute that
the (n; n + i)th entry of QQk , for 06i6k , is given by
(QQk )n; n+i = Qn; n+i−1
− Qn; n+i ( + + n − k + i)
(k) (n + i + 1) (k+1) (n + i)
(k) (n + i) (k+1) (n + i + 1)
(−1)i+k+1 nj=n−k+i ( + + j )
=
(k) (n + i) (k+1) (n + i + 1)
Q
×[i−1 (n + + i − 1) (k+1) (n + i + 1) + i (n + + i) (k+1) (n + i)];
(4.42)
where, in the last equality, we have used Eq. (4.32) and, by convention, we put −1 (n) = 0. Thus, to
establish that Eq. (4.40) holds for QQk , one just needs to show that the term between the brackets
[ : : : ] in the last equality of Eq. (4.42) is
[ : : : ] = (k) (n + i)i(k+1) (n + + i);
(4.43)
where i(k+1) (n) denotes the (k + 1) × (k + 1) determinant obtained by replacing k by k + 1 in the
de nition of i (n), see Eqs. (4.32), (4.16) – (4.18).
We now establish Eq. (4.43) for 16i6k . The case i = 0 is trivial remembering that −1 (n + −
1) = 0. Consider the (k + 2) × (k + 1) matrix
(l−1)
(f˜n+i−k−2+j ) 16j6k+2 ;
16l6k+1
with j and l denoting respectively the line and the column indices. We take V (resp. W ) to be the
subspace of Rk+2 generated by the rst k columns (resp. all the columns) of this matrix. Applying
Lemma 4.4 to this choice of V and W , if we pick in (4.21) (i; j; l) ≡ (1; k + 2 − i; k + 2), we get
that
p2::: k+1 1::: k+2−i:::
1:::k+1 :
[
[
[
k+2 = p1::: k+2−i:::
k+1 2::: k+2 + p2:::k+2−i:::k+2
(4.44)
Using the de nitions (4.16) – (4.18) and (4.32) of the various determinants involved in Eq. (4.43),
one checks easily that this relation amounts precisely to Eq. (4.44), i.e. it is identically satis ed.
This concludes the proof of Lemma 4.5.
It follows immediately from Lemma 4.5, that, in terms of matrices, the operator Q takes the form
Q = SV −1 ;
(4.45)
with
S=
k
X
i+k
(−1)
i=0
V = 0 ( + I ):
T
i
k−1
Y
( + ( − j )I )i ();
(4.46)
j=i
(4.47)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
By combining the results of this section we obtain
Theorem 3. Let be a positive integer and be arbitrary. Then; the (bi-in nite) tridiagonal
matrix which is obtained after k elementary Darboux transformations starting from L( ; ) as in
Eq. (4:1); with 16k6 ; is bispectral.
Proof. The result follows immediately by applying Theorem 1 to the factorization (4.10), with
; R; S and V as in Eqs. (4.37), (4.38), (4.46) and (4.47), using that = b(L( ; )k ) = z k . Given
any family of functions {fn (z )}n ∈ Z belonging to the two-dimensional space of bispectral functions
of L( ; ), which satisfy Eqs. (2.2) and (2.4), we get that the new functions
f˜n = (Pf)n ;
(4.48)
with P as in Eq. (4.33), satisfy
B(k) f˜n = n(k) f˜n ;
(4.49)
with
B(k) = b(R)b(S )z −k ;
n(k) = 0 (n + + 1)0 (n + )
k−1
Y
(n + + − j );
(4.50)
j=0
0 (n) as in Eq. (4.32) and b the bispectral anti-isomorphism de ned in Eqs. (2.16), (2.17), (2:17′ ).
This establishes Theorem 3.
The functions f˜n (z ) that we obtain in Eq. (4.48) by taking f = {fn (z )}n ∈ Z to be the Laguerre
functions introduced in Eq. (2.13), will be called the Krall–Laguerre functions. We remind the
reader (see Remark 4:2) that the matrix P in Eq. (4.48), involved in the Darboux factorization
(4.10), depends on k -free parameters
0 ; : : : ;
k−1 (equivalent to 0 ; : : : ; k−1 in Eq. (4.31)), when
we normalize the functions f( j) in Eq. (4.5) as in Eq. (4.13). Thus, the Krall–Laguerre functions
f˜n (z ) depend on the parameters ( ; ; 0 ; : : : ; k−1 ). When = 0
www.elsevier.nl/locate/cam
Some functions that generalize the Krall–Laguerre polynomials
F. Alberto Grunbauma; ∗; 1 , Luc Haineb; 2 , Emil Horozovc; 3
a
b
Department of Mathematics, University of California, Berkeley, CA 94720, USA
Department of Mathematics, Universite Catholique de Louvain, 1348 Louvain-la-Neuve, Belgium
c
Department of Mathematics and Informatics, So a University, So a 1126, Bulgaria
Received 29 December 1998
Abstract
Let L( ) be the (semi-in nite) tridiagonal matrix associated with the three-term recursion relation satis ed by the
1
Laguerre polynomials, with weight function ( +1)
z e−z ; ¿ − 1, on the interval [0; ∞[. We show that, when is a
positive integer, by performing at most successive Darboux
Pk transformations from L( ), we obtain orthogonal polynomials
1
on [0; ∞[ with ‘weight distribution’ ( −k+1)
z −k e−z + j=1 sj (k−j) (z), with 16k6 . We prove that, as a consequence
of the rational character of the Darboux factorization, these polynomials are eigenfunctions of a ( nite order) di erential
operator. Our construction calls for a natural bi-in nite extension of these results with polynomials replaced by functions,
c 1999 Elsevier Science B.V. All rights reserved.
of which the semi-in nite case is a limiting situation.
1. Introduction
The bispectral problem, as originally posed and solved by Duistermaat and Grunbaum [8], consists
2
in nding all Schrodinger operators L = ddx2 + V (x), for which some families of eigenfunctions f(x; z ),
satisfying Lf = zf, are also eigenfunctions of a di erential operator of arbitrary (but xed) order
in the spectral variable z; Bf = (x)f. The complete solution of this problem revealed its intimate
connection with the theory of Darboux transformations and integrable systems. In a nutshell, all
solutions of the problem can be obtained by means of repeated application of the Darboux process
to some of its basic solutions. By a ‘basic’ solution, we mean a solution for which the bispectral
operator B is of lowest possible order, in this case of order 2.
∗
Corresponding author.
Supported in part by NSF Grant # DMS94-00097 and by AFOSR under Contract FDF49620-96-1-0127.
2
A Research Associate of the Belgian National Fund for Scienti c Research.
3
Supported in part by Grant MM-523 of Bulgarian Ministry of Education.
1
c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ - see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 9 ) 0 0 0 6 9 - 2
272
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
In [13] some of us formulated a discrete–continuous version of the original problem, where the
Schrodinger operator is replaced by a doubly in nite tridiagonal matrix
..
.
L=
..
.
a−2
..
.
b−1
a−1
1
b0
a0
1
b1
a1
1
b2
a2
1
b3
..
.
1
..
.
..
.
;
(1.1)
with the convention that b1 is the (0; 0)th entry of L. More precisely, the problem consists in
determining all bi-in nite tridiagonal matrices L, such that at least one family of eigenfunctions
fn (z ); n ∈ Z, given by
an fn−1 (z ) + bn+1 fn (z ) + fn+1 (z ) = zfn (z );
(1.2)
is also a family of eigenfunctions of a di erential operator of order m (with coecients independent
of n ∈ Z):
m
X
d
d i fn (z )
B z;
= n fn (z ):
fn (z ) ≡
hi (z )
dz
d zi
i=0
(1.3)
The ‘basic’ solutions correspond again to the case m = 2. They are obtained by shifting n to
n + in the three-term recursion relation satis ed by either the Hermite, the Laguerre, the Jacobi
or the (lesser known) Bessel polynomials, letting n run over all integers, see [13,16]. Observe
that the ‘associated polynomials’, see [1,2], satisfy the corresponding three-term recursion relation.
However, as soon as 6= 0, the family of common solutions to Eqs. (1.2) and (1.3) is not given by
the associated polynomials, but rather by functions which can be speci ed in terms of an arbitrary
solution of Gauss’ hypergeometric equation or some of its con
uences, see [13]. It is only when =0,
that one can put f−1 (z ) = 0 and f0 (z ) = 1 and replace the matrix L in Eq. (1.1) by the semi-in nite
matrix obtained by chopping all the columns to the left of b1 and all the rows above b1 . In this case
the fn ’s become the Hermite, the Laguerre, the Jacobi and the Bessel polynomials. This gives back
the classical result of Bochner [6], characterizing the classical orthogonal polynomials as the only
families of orthogonal polynomials, which are eigenfunctions of a second order di erential operator.
The more general problem of determining all orthogonal polynomials which are eigenfunctions of
a di erential operator of arbitrary order was formulated by Krall [24], back in 1938. He found that
the operator had to be of even order and, in [25], he gave the complete solution for m = 4. The new
orthogonal polynomials that he discovered are now coined under the name of ‘Krall polynomials’.
The Krall polynomials are strongly related to some instances of the classical orthogonal polynomials and, in [12], Grunbaum and Haine showed that they can be obtained from some instances
of the Laguerre and the Jacobi polynomials by one or two applications of the (matrix) Darboux
process at the end points of their interval of orthogonality. At this point, the reader will have noticed the absence of the Hermite and the Bessel polynomials as ‘building blocks’ for corresponding
Krall polynomials. The ‘mystery’ was partially elucidated in [15], by showing how to produce out
of the Hermite and any instance of the Bessel polynomials, a pentadiagonal matrix with a family
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
273
of polynomial eigenfunctions that satis es a xed di erential equation of order 4. The same construction can be performed out of any instance of the Jacobi and the Laguerre polynomials [14];
the Krall polynomials are distinguished as the special cases when the above pentadiagonal matrix
is the square of a tridiagonal matrix. This never happens in the case of the Hermite and the Bessel
polynomials.
In this paper we show that repeated application of the Darboux process starting from the semiin nite matrix L( ), associated with the three-term recursion relation satis ed by the Laguerre polynomials, for positive integer values of its parameter , leads to orthogonal polynomials which are
eigenfunctions of a ( nite order) di erential operator. Actually, we will see that it is natural to
enlarge the problem to the two-parameters bi-in nite extension L( ; ) of the matrix L( ) mentioned
above, which reduces to it when = 0. When =
6 0, we obtain in this way functions which provide
higher order instances of solutions of the discrete-continuous version of the bispectral problem, and
which have never appeared before, even in the case of order 4.
One of the key new ideas, that has been introduced in the area of the bispectral problem in
the last few years, is due to Wilson [31,32], who raised the original question at the level of a
commutative algebra of di erential operators, that is at the level of the ‘common eigenfunctions’ of
the algebra. This led him to introduce the seminal idea of the bispectral involution, which amounts
to interchanging the role of the ‘space’ and the ‘spectral’ variables in these eigenfunctions. In
this way, he obtained a complete description of all (maximal) rank one commutative algebras of
bispectral di erential operators. Recently, his ideas were further developed in the works of Bakalov
et al. [3–5] and Kasman and Rothstein [17], aiming at obtaining further examples of higher order
rank commutative algebras of bispectral di erential operators, the rst examples of which appeared
already in the original work of Duistermaat and Grunbaum [8]. Although the methods of [8,31]
appeared to be very di erent, these works allowed to see them as part of a general theory, by
introducing the concept of a bispectral Darboux transformation. For the study of some new and
intriguing examples of bispectral ordinary di erential operators, of a di erent nature than those in
[8], we refer the reader to [4,5,11].
An important message of this paper is to show that the bi-in nite extension of Bochner’s original
result [6], obtained in [13], is the natural context in which Wilson’s ideas can be adapted to the
discrete-continuous version of the bispectral problem. Indeed, only bi-in nite tridiagonal matrices
possess a two-dimensional kernel, from which bispectral Darboux transformations can be performed.
In our context, Wilson’s bispectral involution becomes an anti-isomorphism from an algebra of
matrices to an algebra of di erential operators. This anti-isomorphism is given explicitly by the
three-term recursion relation, the di erentiation formula and the second-order equation which are
satis ed by the two-dimensional space of ‘bispectral functions’, corresponding to the ‘basic’ solutions
of the problem.
Section 2 describes the two-dimensional space of common eigenfunctions to Eqs. (1.2) and (1.3),
when the matrix L in Eq. (1.1) is the bi-in nite Laguerre matrix L( ; ). The use of one Darboux
transformation is presented in detail in Section 3, where we obtain results that had been previously
obtained by other methods when = 0. This will prepare the reader for the more technical Section 4
which is concerned with the iteration of the Darboux process. When =0 and the number of Darboux
transformations is less than +1; ¿ 0, we obtain in this way orthogonal polynomials which extend
Koornwinder’s generalized Laguerre polynomials [20] (see also [18]), with weight distributions
involving not only the delta function, but also its derivatives. If we start the Darboux process at a
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
positive integer value of , these orthogonal polynomials are eigenfunctions of a di erential operator
and, to the best of our knowledge, this result is new. When 6= 0, there is a family of functions
which solve simultaneously Eqs. (1.2) and (1.3) and become polynomials only when = 0. We
propose to call these functions the Krall–Laguerre functions. Finally, in Section 5, we illustrate the
theory by considering the simplest example which in the limit → 0 leads to orthogonal polynomials
with weight distribution involving not only the delta-function but also its rst derivative.
2. The bi-in nite Laguerre matrix and its associated bispectral triple
The bi-in nite Laguerre matrix L( ; ) (in short, L) is a tridiagonal matrix as in Eq. (1.1) which
is obtained by shifting n to n + in the coecients of the standard recursion relation which de nes
the (monic) Laguerre polynomials, with the understanding that n runs over all integers:
an = (n + )(n + + );
bn = 2(n + ) + − 1:
(2.1)
In the sequel, we shall denote by L( ) the semi-in nite matrix which is obtained by chopping all the
columns to the left of b1 and all the rows above b1 in L( ; 0), and de nes the three-term recursion
relation satis ed by the usual (monic) Laguerre polynomials.
It is shown in [13] (see also [16]) that there is a two-dimensional space of functions {fn (z )}n ∈ Z
satisfying the following three properties:
(i) {fn (z )} satisfy the three-term recursion relation
zf = L( ; )f;
(2.2)
with f ≡ (: : : ; f−1 ; f0 ; f1 ; : : :)T ;
(ii) {fn (z )} satisfy a di erentiation formula
A z;
d
f = Mf;
dz
(2.3)
with A a rst-order di erential operator and M a (bi-in nite) tridiagonal matrix;
(iii) {fn (z )} are eigenfunctions of a second-order di erential operator
B z;
d
f = f;
dz
(2.4)
with the diagonal matrix of eigenvalues of B, = diag(: : : ; −1 ; 0 ; 1 ; : : :).
The data listed in (i) – (iii) can be described as follows. Pick w(z ) an arbitrary solution of the
equation
zw′′ (z ) + ( + 1 − z )w′ (z ) + w(z ) = 0;
(2.5)
which is obtained from Gauss’ hypergeometric equation by con
uence. De ne f0 (z ) and f1 (z ) by
f0 (z ) = w(z );
(2.6)
f1 (z ) = (z − − 1 − )w(z ) − zw′ (z ):
(2.7)
and
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
275
Then, the family of functions {fn (z )}n ∈ Z de ned by the three-term recursion relation (2.2) with
f0 (z ) and f1 (z ) as in Eqs. (2.6) and (2.7), satis es automatically Eqs. (2.3) and (2.4) with
d
A=z ;
dz
(2.8)
Mn; n+1 = 0; Mn; n = n + ; Mn; n−1 = (n + )(n + + );
d2
d
+ (z − − 1) ;
2
dz
dz
n = n + :
B = −z
(2.9)
One can always pick [9, 6.3(2)]
w(z ) =1 F1 (−; + 1; z );
(2.10)
to be a solution of Eq. (2.5), with 1 F1 (a; c; z ) denoting the hypergeometric series
1 F1 (a; c; z ) =
∞
X
(a)n
n=0
n!(c)n
zn :
(2.11)
Here, as well as in the rest of the paper, (x)n , n ∈ Z, is the shifted factorial (Pochammer notation):
(x + n)
(x)0 = 1; (x)n =
= (x + n − 1)(x)n−1 :
(2.12)
(x )
It follows from [9, 6.4(2),6.4(9)] that the solution of the three-term recursion relation (2.2) with
initial conditions given by f0 (z ) and f1 (z ) as in Eqs. (2.6) and (2.7), and w(z ) as in Eq. (2.10), is
given by
fn (z ) = (−1)n ( + 1 + )n 1 F1 (−n − ; + 1; z ):
(2.13)
Notice that, when = 0, the functions fn (z ), n¿0, reduce precisely to the Laguerre polynomials,
normalized to be monic, see [9, 10.12(14)]. For this reason, we shall call the functions fn (z )
de ned in Eq. (2.13) the Laguerre functions. They are eigenfunctions of the second order di erential
operator B de ned in Eq. (2.9), which generalizes the standard second order di erential equation
satis ed by the Laguerre polynomials, and they will play a basic role in what follows. As emphasized
in the introduction, we notice that, although the three-term recursion relation satis ed by the functions
fn (z ) is the same as the one satis ed by the associated Laguerre polynomials studied in [2], when
6= 0, the fn (z ) are not polynomials. The ‘associated polynomials’ satisfy Eq. (2.2) but, for =
6 0,
do not satisfy Eq. (2.4).
We shall denote by
B = hL; M; i;
(2.14)
the subalgebra of the algebra of nite band bi-in nite matrices generated by the matrices L, M ,
appearing on the right-hand side of Eqs. (2.2) – (2.4). Similarly
B′ = hz; A; Bi;
(2.15)
will denote the subalgebra of the algebra of di erential operators generated by z , A, B. Formulas
(2.2) – (2.4) serve to de ne an anti-isomorphism
b : B → B′
(2.16)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
between these two algebras, i.e. it is given on the generators by
b(L) = z;
b(M ) = A
and
b() = B:
(2.17)
More precisely, any monomial Li M j k in B, i; j; k¿0, acting on the original space of bispectral
functions {fn }n ∈ Z , gives
Li M j k f = Bk Aj z i f;
i.e.
b(Li M j k ) = b(k )b(M j )b(Li ) = Bk Aj z i :
(2.17′ )
The triple (B; B′ ; b) provides an instance of the notion of a bispectral triple which was introduced
in [4]. To explain the terminology, we need to introduce the commutative subalgebras K ⊂ B and
K′ ⊂ B′ generated respectively by and z . We shall refer to these subalgebras as the ‘algebras
of functions’. Their images by b and b−1 will be denoted by A′ and A respectively and provide
obvious bispectral operators. In order to de ne the notion of Darboux transformation in this abstract
setting, we shall denote with a bar the elds of quotients of K, K′ , A and A′ . Obviously, b
′
′
extends to isomorphisms K → A and A → K . We now reproduce from [15] the main tool that
we need to produce out of L( ; ) new non-trivial bispectral operators by means of the Darboux
transformation, see also [4].
Theorem 1. Let L ∈ A be a constant coecients polynomial in L; which factorizes ‘rationally’
as
L = QP;
(2.18)
in such a way that
Q = SV −1 ;
P = −1 R;
(2.18′ )
with R; S ∈ B and V ∈ K. Then the Darboux transform of L given by
˜ = PQ;
L
(2.19)
is a bispectral operator. More precisely; de ning ≡ b(L) ∈ K′ and f˜ ≡ Pf; with f satisfying
Eqs. (2:2)–(2:4) above; we have
˜
˜ f˜ = f;
L
(2.20)
˜
B˜ f˜ = V f;
(2.21)
B˜ = b(R)b(S )−1 :
(2.22)
with
Proof. Eq. (2.20) follows immediately from the de nitions. Let
Pb ≡ b(R)
and
Qb ≡ b(S )−1 :
(2.23)
′
Clearly, using the anti-isomorphism introduced in Eqs. (2.16), (2.17), (2:17 ),
f˜ = Pf = −1 Pb f:
(2.24)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
277
Since B has no zero divisors, Eqs. (2.18), (2.18′ ) imply
V = RL−1 S; L−1 ∈ A:
Applying the anti-isomorphism b, we obtain
b(V ) = b(S )−1 b(R):
(2.25)
From Eqs. (2.23) – (2.25), we have
˜
Vf = Qb Pb f = Qb f;
and thus, using Eq. (2.18) and this last relation, we deduce that
˜
f = −1 Qf˜ = V −1 Qb f:
(2.26)
This equation combined with Eqs. (2.23) and (2.24) gives Eq. (2.21), which completes the proof
of the theorem.
Remark. Observe that B˜ in Eq. (2:22) is a Darboux transformation of b(V ) in Eq. (2:25). Since
V ∈ K; V is a polynomial in ; and thus b(V ) is a polynomial in B. This shows that the
new bispectral operator B˜ is in fact obtained as a Darboux transform of a constant coecients
polynomial in the original (second order) bispectral operator B.
Notice that Theorem 1 leaves open the question of when a ‘rational’ factorization of the form
(2.18), (2.18′ ) can be performed. Our aim in the next two sections is to show that, for any positive
integer , the successive powers L( ; )k , k = 1; 2; : : : ; , of the doubly in nite Laguerre matrix admit
˜ in Eq. (2.19) is again the
such a factorization with the additional property that the new operator L
−1
˜
k th-power of a tridiagonal matrix. The functions fn = (Pf)n = Pb fn , obtained from the Laguerre
functions fn in Eq. (2.13), are thus solutions of the discrete-continuous version of the bispectral
problem (1.2), (1.3), with a bispectral operator B of order m ¿ 2. We shall refer to them as the
Krall–Laguerre functions. When = 0, these functions become polynomials which generalize the
Krall–Laguerre polynomials. For a precise de nition, see Theorem 3 in Section 4.
3. One step of the Darboux process
In this section we rst perform a standard Darboux transformation on the doubly in nite Laguerre
matrix L( ; ), when ¿ 0 and is arbitrary. In the limit = 0, we obtain in this way orthogonal
1
z −1 e−z +
(z ), where
is the free parameter
polynomials on the interval [0; ∞[ for the measure ( )
of the Darboux transformation. These polynomials (as well as an extension to the Jacobi case) were
found by Koornwinder [20]. In particular, he proved that, in general, they satisfy a second-order
di erential equation with coecients depending on n. In the special case when is a positive
integer, we show in Section 3.2 that the Darboux factorization of L( ; ) can be recast in the form
(2.18), (2.18’) called for in Theorem 1, and as a consequence, the Darboux transform of L( ; )
is again bispectral, in the sense of Eqs. (1.2) and (1.3). When = 0 and = 1, we get back the
classical Krall–Laguerre polynomials, discovered by Krall [25]. When ¿ 1, this result has been
obtained (using other methods) by Littlejohn [29] when = 2, by Krall and Littlejohn [23] when
= 3, and by Koekoek and Koekoek [18] for an arbitrary positive integer . The case 6= 0 appears
here for the rst time and leads to ‘Krall–Laguerre functions’.
278
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3.1. Factorizing the bi-in nite Laguerre matrix
We remind the reader that the standard Darboux transformation (referred to in the sequel as
elementary Darboux transformation) of a bi-in nite tridiagonal matrix L [30] starts by factorizing L
as
L = QP;
(3.1)
with the two factors Q and P denoting, respectively, upper and lower triangular matrices acting on
a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T as follows:
(Qh)n = x n+1 hn + hn+1 ;
(3.2)
(Ph)n = yn hn−1 + hn :
(3.3)
The most general factorization of L, in the form (3.1), is obtained by picking an arbitrary element
f ∈ ker L, so that the matrices P and Q are given by
fn
hn−1 ;
(3.4)
(Ph)n = hn −
fn−1
fn−1
hn + hn+1 :
(3.5)
fn
Since the kernel of L is two-dimensional and only the ratios of the fn ’s are involved, this factorization depends (projectively) on one free parameter. The Darboux transformation L˜ of L is obtained
by exchanging the order of the factors in Eq. (3.1)
L˜ = PQ:
(3.6)
(Qh)n = −an
Explicitly, the entries a˜n and b˜n of the new tridiagonal matrix L˜ are given by
fn−2 fn
a˜n = an−1 2 ;
fn−1
(3.7)
fn−1
fn
−
:
(3.8)
b˜n = bn +
fn−1 fn−2
As observed in [12], the above construction still makes sense in the case of semi-in nite matrices,
that is if one chops all the columns to the left of the (0; 0)th entry and all the rows above the (0; 0)th
entry of the matrices L, P and Q. The crucial observation is that, in this limit, the upper–lower
factorization (3.1) of L, still contains a free parameter, which can be picked to be x1 . This would
no longer be the case, if instead we had chosen to perform a lower–upper factorization of L, in
which case all the entries of P and Q are uniquely speci ed.
Thus, in order to perform the (most general) factorization of the form (3.1) for the bi-in nite
Laguerre matrix L( ; ), we need a description of its two-dimensional kernel. Introducing the lower
shift matrix
(T −1 h)n = hn−1 ;
(3.9)
and the matrix C de ned by
(Ch)n = (n + 1 + )hn + hn+1 ;
(3.10)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
279
one checks easily that
L( ; ) = T −1 C (C + I );
(3.11)
with I the identity matrix. From this important observation, since the matrices C and C + I commute,
we deduce immediately the next result.
Lemma 3.1. Let ¿ 0. The kernel of L( ; ) is generated by the vectors = (: : : ; −1 ; 0 ; 1 ; : : :)T
and = (: : : ; −1 ; 0 ; 1 ; : : :)T ; where
n = (−1)n (1 + )n ;
n
(3.12)
= (−1)n (1 + + )n ;
(3.13)
and (x)n denotes the shifted factorial.
Corollary 3.2. The most general factorization of the bi-in nite Laguerre matrix L( ; ) in the
form (3:1)–(3:3) is given by
an−1
n +
n
;
xn =
;
(3.14)
yn = −
n−1 +
n−1
yn−1
with
an arbitrary free parameter; and n ;
n;
an as in Eqs. (3:12); (3:13) and (2:1).
Proof. From Lemma 3.1, we have that
f ∈ ker L( ; ) ⇔ f = +
;
with
an arbitrary free parameter (we allow
= ∞; i.e. f = ), which using Eqs. (3.4) and(3.5)
gives Eq. (3.14) and establishes the corollary.
One checks easily that in the limit → 0; y0 → 0 and x1 → = (1 +
); and thus one obtains a
factorization of the semi-in nite Laguerre matrix, where x1 is equivalent to the free parameter
.
The Darboux transform L˜ (3.6) de nes new orthogonal polynomials qn −1;
given by
qn −1;
= (Pp )n
= pn +
n! +
( + 1)n
p ;
(n − 1)! +
( + 1)n−1 n−1
1
z e−z on [0; ∞[;
where pn (z ) denote the (monic) Laguerre polynomials with weight function ( +1)
when ¿ − 1. Using the standard formulas (see, for instance, [9, 10.12(15),10.12(16)])
1 d −1
d
=
pn−1
pn −1 ;
and pn = 1 −
pn
n dz
dz
we get that
d −1
( )n
qn −1;
= pn −1 +
p ;
n! +
n( + 1)n−1 d z n
which precisely agrees with the formula given in [18,20] for Koornwinder’s generalized Laguerre
1
polynomials (normalized to be monic) with weight function ( )
z −1 e−z +
(z ) on the interval
[0; ∞[; when ¿ 0 and
¿0.
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3.2. Factorizing in the algebra B; when is a positive integer
In the special case when is a positive integer, it is possible to achieve the form of the factorization (2.18), (2:18′ ); called for in Theorem 1. For this, it will be convenient to use another basis
of the algebra B = hL; M; i introduced in Eq. (2.14), namely
B = hN; T; i;
(3.15)
where N denotes the strictly lower part of the bi-in nite Laguerre matrix L = L( ; ):
(Nh)n = an hn−1 = (n + )(n + + )hn−1 ;
(3.16)
and T is the upper shift matrix
(Th)n = hn+1 :
(3.17)
One checks easily that the change of basis is given by
N = M − ;
T = L − − M − ( + 1)I:
(3.18)
The point of this change of basis is that now the three generators N; T; are lower, upper and
diagonal matrices, respectively, with only one non-zero diagonal. They give the most convenient
way to factorize a tridiagonal matrix. Using these new generators for B; we have
Lemma 3.3. Let be a positive integer. Then; the Darboux factorization (3:14) of L( ; ) can
be recast into the form called for in Theorem 1:
L( ; ) = QP ≡ (SV −1 )(−1 R);
(3.19)
with R; S ∈ B given by
R = ( + I )() + N( + 2I );
(3.20)
S = ( + I )() + T( + I );
(3.21)
= ()( + I );
(3.22)
and
V = ( + I );
where () a polynomial of degree in
( ) = I +
−1
Y
( + jI );
j=0
=
:
(1 + )
(3.23)
Proof. Using that is a positive integer and the de nition of an in Eq. (2.1), the entries yn and x n
(3.14) of the factors Q and P in the Darboux factorization (3.1) become rational functions of n:
(n + )(n + + 1)
(n − 1 + + )(n + −1)
yn =
;
xn =
;
(3.24)
(n + )
(n + )
with (n) a polynomial of degree in n:
(n) = 1 + (n) ;
=
= (1 + ) :
(3.25)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
281
Since
(Ph)n = hn + yn hn−1
1
=
((n + )hn + (n + )(n + + 1)hn−1 );
(n + )
(3.26)
remembering that is the diagonal matrix = diag(n ); n = n + ; with 0 = at the (0; 0)th entry,
we can rewrite P as follows
P = ()−1 [() + T −1 ( + I )( + 2I )]:
(3.27)
The (lower) shift matrix T −1 does not belong to the algebra B; but we observe that, by the de nition
of the an ’s (2.1), the matrix N ∈ B (3.16) can be written as
N = T −1 ( + I )( + (1 + )I )
= ( + I )T −1 ( + I ):
(3.28)
Therefore,
P = ()−1 [() + ( + I )−1 N( + 2I )]
= ()−1 ( + I )−1 [( + I )() + N( + 2I )]:
(3.29)
Similarly,
(Qh)n = x n+1 hn + hn+1
= (n + + )
(n + )
hn + hn+1 ;
(n + + 1)
(3.30)
and thus,
Q = ( + I )()( + I )−1 + T
= [( + I )() + T( + I )]( + I )−1 :
(3.31)
Since the (upper) shift matrix T belongs to B; the factor between the brackets on the right-hand
side of the last equality belongs automatically to B. Combining Eqs. (3.25), (3.29) and (3.31) gives
the result announced in the lemma, which concludes the proof.
Notice that in this case, both and V in Eq. (3.22) belong to the algebra of ‘functions’ K
(i.e. the constant coecients polynomials in ), and thus the product V ∈ K. Since we already
observed that R and S in Eqs. (3.20) and(3.21) belong to the algebra B; using Theorem 1 with
= b(L( ; )) = z; we deduce immediately
Corollary 3.4. Let be a positive integer. The Darboux transform L˜ = PQ of L( ; ); with P; Q
in Eq. (3:19) is again bispectral. Explicitly; de ning
f˜n = (Pf)n ;
with fn as in Eq. (2:13); we have
L˜f˜n = z f˜n ;
B˜ f˜n = (n + + )(n + )(n + + 1)f˜n ;
(3.32)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
with
B˜ = b(R)b(S )z −1 ;
(3.33)
where b : B → B′ is the bispectral anti-isomorphism de ned in Eqs. (2:16); (2:17); (2:17′ ).
To see that these formulas are actually constructive, we compute from Eqs. (2.17) and (3.18)
b() = B; b(N ) = A − B;
b(T ) = z − B − A − ( + 1);
(3.34)
and then, using the fact that b in Eq. (2.16) is an anti-isomorphism, we get from Eqs. (3.20), (3.21)
that
b(R) = (B)(B + ) + (B + 2)(A − B);
(3.35)
b(S ) = (B)(B + ) + (B + 1)(z − B − A − ( + 1)):
Recalling the remark following Theorem 1, B˜ is a Darboux transform of b(V ). Since (V )n =
(n + + )(n + )(n + + 1) is a polynomial of degree 2 + 1 in n + ; both b(V ) and B˜ in Eq.
(3.33) are operators of order 2(2 + 1). This order is larger than the order of the operator produced
in [18]. The issue of getting the lowest possible order will be discussed in Section 5.
4. Iterating the Darboux process
In this section we rst describe the result of k iterations of the Darboux process starting from
the bi-in nite Laguerre matrix L( ; ); ¿ 0; k ¡ + 1. We compute the moment functional for the
resulting orthogonal polynomial sequence, when = 0. Then, in Section 4.2, when is a positive
integer and the number of elementary Darboux transformations is less than or equal to ; we show
that the process leads to functions which solve simultaneously Eqs. (1.2) and (1.3). In the limit =0;
these functions become orthogonal polynomials which are eigenfunctions of a di erential operator
and generalize the Krall–Laguerre polynomials; for this reason, we call them the Krall–Laguerre
functions.
4.1. Extending Koornwinder’s generalized Laguerre polynomials
In order to compute the e ect of k successive elementary Darboux transformations starting from
L0 ≡ L( ; )
L0 = Q0 P0 y L1 = P0 Q0 = Q1 P1 y : : :
y Lk−1 = Pk−2 Qk−2 = Qk−1 Pk−1 y Lk = Pk−1 Qk−1 ;
(4.1)
we need the following lemma, which follows easily from the representation (3.11) of L( ; ) and
generalizes Lemma 3.1:
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
j)
Lemma 4.1. For j¿0; the vectors ( j) =(: : : ; (−1
; (0j) ; (1j) ; : : :)T and
with entries
(nj) = (−1)n+j
( j)
n
= (−1)n+j
( j)
=(: : : ;
( j)
−1 ;
( j)
0 ;
283
( j)
T
1 ; : : :) ;
(1 + )n+j
;
j !(1 − )j
(4.2)
(1 + + )n+j
;
j !(1 + )j
(4.3)
satisfy
L( ; ) ( j) = ( j−1)
and
L( ; )
with the natural interpretation (−1) =
( j)
=
(−1)
( j−1)
;
(4.4)
= 0.
Let 0 ; : : : ; k−1 ;
0 ; : : : ;
k−1 denote free parameters and, for 06j6k − 1, de ne
f
( j)
=
j
X
( l ( j−l) +
l
( j−l)
):
(4.5)
l=0
Since, from Lemma 4.1,
L0 f( j) = f( j−1) ;
(4.6)
one shows inductively that, for 06j6k − 1,
ker Pj = (one-dimensional) span of Pj−1 Pj−2 : : : P0 f( j) :
(4.7)
When j = 0, this formula is interpreted as ker P0 = f(0) , reproducing the result given in Lemma 3.1.
Thus, the product matrix P
P = Pk−1 Pk−2 : : : P0 ;
(4.8)
has a kernel generated by f(0) ; f(1) ; : : : ; f(k−1) , and therefore it can be written as the ratio of two
Casorati determinants (using the action on a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T ):
(0)
(k−1)
fn−k
: : : fn−k
hn−k
..
..
.
..
.
.
(k−1)
f(0)
:
:
:
f
h
n
n
(Ph)n = n (0)
(k−1)
: : : fn−k
fn−k
.
..
.
.
.
(0)
(k−1)
fn−1
: : : fn−1
:
(4.9)
Another way to put this is to say that the sequence of elementary Darboux transformations (4.1)
can be performed in one shot by factorizing
L ≡ Lk0 = QP;
(4.10)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
with P as in Eq. (4.9) and Q a uniquely determined upper triangular matrix with k + 1 diagonals,
with the entries on the top diagonal normalized to be one. Then, the Darboux transform is
˜ = PQ = (Lk )k ;
L
(4.11)
with Lk the tridiagonal matrix obtained at the end of the chain of the elementary Darboux transformations (4.1). Indeed, since
PL0 = Lk P;
we deduce that
˜
(Lk )k = P (L0 )k P −1 = PQPP −1 = L:
Remark 4.2. From Eq: (4:9) we see that two sequences of parameters ( l ;
l ) and ( l′ ;
′l ); 06l6k−
1; lead to the same factorization (4.10) as long as the corresponding functions f( j) in Eq: (4:5);
06j6k − 1; generate the same
ag of subspaces hf(0) i ⊂hf(0) ; f(1) i ⊂ · · · ⊂hf(0) ; f(1) ; : : : ; f(k−1) i.
In the basis given by ( j) and ( j) in Eqs: (4:2); (4:3); this
ag is represented by the k × 2k matrix
0
0
:::
1
( ;
) ≡ ...
0
0
..
.
..
.
k−1
1
:::
0
..
.
..
.
1
and can be normalized as
0
1
..
.
0
0
k−1
0
:::
0
0
..
.
..
.
1
:::
..
0
..
.
.
1
;
0
(4.12)
0
(I;
) or ( ; I ) ≡ (
−1 ; I );
(4.13)
depending on whether 0 6= 0 or
0 6= 0; with I the identity matrix. Thus; there are in fact only
k free parameters
0 ; : : : ;
k−1 involved in the factorization (4.10); which can be thought of as the
new free parameters involved in the successive elementary Darboux transformations (4.1).
Our aim now is to compute explicitly the entries of the new tridiagonal matrix Lk in Eq. (4.11).
For this, we need to introduce a few notations that will be repeatedly used in this section.
We can write fn( j) in Eq. (4.5) as
( j)
fn( j) = (1 + )n f˜n ;
(4.14)
with
( j)
f˜n
j
X
(−1) j−l
n
= (−1)
l=0
"
#
(n + + 1)j−l
(1 + + )n+j−l
l
:
+
l
( j − l)!
(1 − )j−l
(1 + )n (1 + )j−l
(4.15)
Now we come to some expressions which will play a crucial role. Section 5 illustrates their usefulness in the simplest case = k = 2. We introduce the determinants
( j)
(k) (n) = det(f˜n−k+i )06i; j6k−1 ;
(4.16)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
˜(0)
fn−k
..
.
(0)
˜
fn−i−1
i (n) = (0)
f˜
n−i+1
.
..
(0)
f˜n
(1)
f˜n−k
..
.
(1)
f˜n−i−1
(1)
f˜n−i+1
..
.
(1)
f˜n
:::
:::
:::
:::
for 16i6k − 1, and put
285
(k−1)
f˜n−k
..
.
(k−1)
f˜n−i−1
;
(k−1)
f˜n−i+1
..
.
(4.17)
(k−1)
f˜n
k (n) = (k) (n + 1):
(4.18)
The following lemma extends Eqs. (3.7) and (3.8) to an arbitrary number of (elementary) Darboux
transformations:
Lemma 4.3. The entries an(k) and bn(k) of the tridiagonal matrix Lk ; obtained from L0 = L( ; ) after
k elementary Darboux transformations; are given by
(k) (n − 1) (k) (n + 1)
;
[ (k) (n)]2
(4.19)
1 (n − 1)
1 ( n )
− (n + − 1) (k)
;
(k)
(n)
(n − 1)
(4.20)
an(k) = (n + )(n + + − k )
bn(k) = bn + (n + )
with bn as in Eq: (2:1).
The following sublemma (whose proof we omit) will be needed to establish Lemma 4.3, as well
as Lemma 4.5 later in this section. Both of these lemmas, whose detailed proofs are given, are used
to establish Theorem 2 and Theorem 3 in this section. They produce very concrete formulas which
will be illustrated in Section 5.
Lemma 4.4. Let V and W be vector spaces of respective codimension 2 and 1 in a vector space E
of dimension k + 2. The (incidence) relation V ⊂ W amounts to the system of quadratic equations
satis ed by the Plucker coordinates p1::: i:::
ˆ jˆ ::: k+2 ; 16i ¡ j6k + 2; (resp. 1::: i:::
ˆ k+2 ; 16i6k + 2) of
V (resp. W ):
1::: i:::
ˆ k+2 p1::: jˆ ::: lˆ::: k+2 − 1 ::: jˆ ::: k+2 p1::: iˆ::: lˆ::: k+2
+1::: lˆ::: k+2 p1::: iˆ::: jˆ::: k+2 = 0;
for all 16i ¡ j ¡ l6k + 2:
(4.21)
Proof. See [10, p. 134, Lemma 2], where the more general case of the incidence relations satis ed
by planes V , W of codimension p and q, p¿q, in a vector space E of dimension n, is discussed.
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
Proof of Lemma 4.3. The proof is done by induction on k . For k = 1, we have that
a(1)
n = an−1
(0)
fn−2
fn(0)
;
(0) 2
[fn−1
]
using Eq: (3:7);
(0)
= an−1
(0)
(1 + )n−2 (1 + )n f˜n−2 f˜n
;
(0)
(1 + )2n−1 [f˜n−1 ]2
by Eq : (4:14);
(1) (n − 1) (1) (n + 1)
;
[ (1) (n)]2
using Eqs: (2:1) and (4:16) with k = 1:
= (n + ) (n + + − 1)
Assume now that formula (4.19) holds for k , we establish it for k + 1. Indeed, since by Eq. (4.7),
the kernel of Lk is spanned by
f = Pk−1 Pk−2 : : : P0 f(k) ;
(4.22)
using Eq. (3.7), we get that
(k) fn−2 fn
an(k+1) = an−1
:
2
fn−1
But, from Eqs. (4.8), (4.9), (4.14), (4.16) and (4.22), we see that
(1 + )n (k+1) (n + 1)
;
(k) (n)
and, therefore, by induction hypothesis,
fn =
(4.23)
(n + )[ (k) (n − 1)]2 (k+1) (n − 1) (k+1) (n + 1)
;
(n − 1 + ) (k) (n − 2) (k) (n)[ (k+1) (n)]2
(k+1) (n − 1) (k+1) (n + 1)
= (n + )(n + + − k − 1)
:
[ (k+1) (n)]2
(k)
an(k+1) = an−1
Similarly, to establish Eq. (4.20), we rst observe that, from Eqs. (3.8) and (4.14), we have that
b(1)
n
(0)
(0)
f˜n−1
f˜n
= bn + (n + ) (0) − (n + − 1) (0) ;
f˜n−1
f˜n−2
with bn as in Eq. (2.1), which using Eqs. (4.16) and (4.18), with k = 1, agrees with Eq. (4.20).
Assume now that formula Eq. (4.20) holds for k , we establish it for k + 1. Indeed, by Eq. (3.8),
we have that
fn−1
fn
−
;
bn(k+1) = bn(k) +
fn−1 fn−2
with fn as in Eq. (4.23). By induction hypothesis, this gives that
(k) (n − 1) (k+1) (n + 1) + 1 (n) (k+1) (n)
(k) (n) (k+1) (n)
(k) (n − 2) (k+1) (n) + 1 (n − 1) (k+1) (n − 1)
−(n + − 1)
:
(k) (n − 1) (k+1) (n − 1)
bn(k+1) = bn + (n + )
(4.24)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
287
Consider now the (k + 2) × (k + 1) matrix
( j−1)
16i6k+2 ;
(f˜n−k−2+i ) 16j6k+1
and the subspaces V (resp. W ) of Rk+2 generated by the rst k columns (resp. all the columns)
of this matrix. Applying Lemma 4.4 to this situation, one checks easily that Eq. (4.21) with i = 1,
j = k + 1 and l = k + 2, amounts to
(k) (n − 1) (k+1) (n + 1) + 1 (n) (k+1) (n) = (k) (n)1(k+1) (n);
with 1(k+1) (n) the (k + 1) × (k + 1) determinant obtained by replacing k by k + 1 into Eq. (4.17).
Thus, Eq. (4.24) coincides with Eq. (4.20), with k replaced by k + 1. This establishes Lemma 4.3.
Remember that, as explained in Section 3, in the case of semi-in nite matrices, the (0; 0)th entry
x1 of Q (3.2) is equivalent to the free parameter of the elementary Darboux transformation. Thus
our construction still makes sense in the limit = 0 and the k free parameters can be thought of to
be the entries x1( j) , 06j6k − 1, of the matrices Qj in Eq. (4.1). The matrix Lk which is obtained at
the end of the chain de nes a new sequence of polynomials q(k) = (q0(k) (z ) = 1; q1(k) (z ); q2(k) (z ); : : :)T
satisfying zq(k) = Lk q(k) . By Favard’s theorem (see, for instance, [7]), as long as the entries an(k) of
Lk are non-zero, there exists a unique (up to a multiplicative constant) moment functional M(k) for
which the sequence {qn(k) (z )}∞
n=0 is an orthogonal polynomial sequence, that is
M(k) [qn(k) qm(k) ] = 0;
for m 6= n
and
M(k) [(qn(k) )2 ] 6= 0:
We remind the reader that the moment functional M corresponding to a sequence of complex
numbers {n }∞
n=0 is a complex valued function de ned on the vector space of all polynomials by
M(z n ) = n ;
n = 0; 1; 2; : : : ;
M(c1 1 (z ) + c2 2 (z )) = c1 M(1 (z )) + c2 M(2 (z ));
for all complex numbers ci and all polynomials i (z ) (i = 1; 2).
The next theorem shows that, iterating the Darboux process starting from the Laguerre polynomials, leads to a natural extension of Koornwinder’s generalized Laguerre polynomials (which
correspond to k = 1).
Theorem 2. The sequence of polynomials {qn(k) (z )}∞
n=0 ; obtained after k successive elementary
Darboux transformations starting from the Laguerre matrix L( ); with k ¡ +1; is an orthogonal
sequence of polynomials with moment functional M(k) given by the weight distribution
k
X
(−1)k−j (k−j)
1
z −k e−z +
rj
(z );
( − k + 1)
(k − j )!
j=1
(4.25)
with
rj =
( − k + 1)k
− ( − k + 1)k−j ;
x1(0) : : : x1( j−1)
(4.26)
where x1(0) ; : : : ; x1(k−1) denote the successive free parameters in the elementary Darboux transformations ((0; 0)th entries of Q0 ; : : : ; Qk−1 ); ( j) (z ) denotes the j th derivative of the delta function and
(a)j is the shifted factorial.
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Proof. Start with any orthogonal polynomial sequence {pn (z )}∞
n=0 for a moment functional M, with
M[z n ] = n and 0 = 1. One just needs to observe that, after one elementary Darboux transformation,
if the new matrix L˜ in Eq. (3.6) satis es the hypothesis of Favard’s theorem (i.e. a˜n 6= 0, n¿1),
˜
the resulting sequence of polynomials {qn (z )}∞
n=0 de ned by L will be orthogonal for the moment
˜
functional M given by
M˜ [z n ] = x1 n−1 ;
M˜ [1] = 1;
n¿1;
(4.27)
with x1 the free parameter ((0; 0)th entry of Q) of the Darboux transformation. Indeed, since q = Pp,
one also has
zp = Qq ⇔ zpn = x n+1 qn + qn+1 :
Applying M˜ to both sides, we obtain for n = 0 that M˜ [z ] = x1 , and inductively, using the classical
formula expressing the orthogonal polynomials in terms of the moments
1
1
−1
pn (z ) = n−1 ...
n−1
1
1
2
..
.
:::
:::
n
z
:::
:::
n
n+1
..
.
;
2n−1
zn
˜ n
with n−1 = det(i+j )i;n−1
j=0 , one deduces that M[z ] = x1 n−1 .
From (4.27) it follows that, after k elementary Darboux transformations (again assuming that at
each step we can apply Favard’s theorem), the resulting polynomials {qn(k) (z )}∞
n=0 will be orthogonal
for the moment functional M(k) de ned by
M(k) [1] = 1;
M(k) [z j ] = x1(k−1) x1(k−2) : : : x1(k−j) ;
(k)
j
M [z ] =
x1(k−1) x1(k−2)
: : : x1(0) j−k ;
16j6k − 1;
j¿k:
By Lemma 4.3 (formula (4.19), with = 0), we see that as long as k ¡ + 1, an(k) 6= 0 for n¿1,
and thus the argument above applies. Since the moments {sn }∞
n=0 of the weight distribution (4.25)
are given by
sk−j
( − k + 1)k−j + rj
=
; 16j6k − 1;
s0
1 + rk
sj
( − k + 1)k
=
j−k ; j¿k;
s0
1 + rk
we see that this distribution de nes the same moment functional as M(k) , provided that we pick the
rj ’s as in Eq. (4.26), which establishes Theorem 2.
4.2. The bispectral property
We now show that, when is a positive integer and k6 , the factorization (4.10) can be put
into the form (2.18), (2:18′ ) needed to apply Theorem 1. As a consequence, the resulting tridiagonal
matrix Lk obtained after k elementary Darboux transformations will be bispectral. The punch line
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
289
of the argument is to exploit in this case the rational character of the Darboux factorization (4.10).
The denominators from the left-hand side and the right-hand side factors are taken out to produce
part of the ‘function’ V ∈ K. Since the upper-shift matrix T is in the algebra B, what remains
to the left is automatically in B. We expand what remains on the right-hand side in terms of the
lower-shift matrix T −1 , then express the various powers of T −1 in terms of N via Eq. (3.28) and
absorb the denominator in . We now proceed with the details of our program.
Put
p i (n ) ≡
i
Y
(n + j ):
(4.28)
j=1
( j)
Since is a positive integer, we can write f˜n in Eq. (4.15) as
( j)
f˜n = (−1)n g( j) (n + );
(4.29)
with
g( j) (n) =
j
X
(−1) j−l
l=0
where
"
#
l
l
pj−l (n) +
p +j−l (n) ;
( j − l)! (1 − )j−l
(1 + )j−l
l =
l = (1 + ) :
(4.30)
(4.31)
It follows from Eqs. (4.28) – (4.30) that the determinants (k) (n) and i (n), 16i6k , de ned in
Eqs. (4.16) – (4.18), become now polynomials in the variable n + . Thus, we shall write
(k) (n) ≡ 0 (n + );
i (n) ≡ i (n + );
16i6k;
(4.32)
( j)
where the i (n), 06i6k , are the determinants obtained by replacing f˜l by (−1)l g( j) (l) respectively
in Eqs. (4.16) – (4.18).
( j)
Remembering the de nition of f˜n in Eq. (4.14), from each of the rst k columns of both
the numerator and the denominator of Eq. (4.9), we can extract a factor (1 + )n−k . In this way,
expanding the numerator of Eq. (4.9) along the last column, we get
(Ph)n =
k
X
(−1)i
i=0
n
Y
( + j )
j=n−i+1
i (n + )
hn−i ;
0 (n + )
(4.33)
with i (n), 06i6k
, de ned as in Eq. (4.32). Here and in the rest of the paper we make the
Q
convention that nj=n+1 (: : :) = 1.
From the previous formula, it is clear that, in terms of matrices, the operator P can be written as
P = 0 ()−1
k
X
i=0
(−1)i T −i
i
Y
( + jI )i ( + iI ):
(4.34)
j=1
From Eq. (3.28) we deduce easily that
Ni =
i−1
Y
j=0
( + ( − j )I )T −i
i
Y
j=1
( + jI );
(4.35)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
and so, we can rewrite P as
P = −1 R;
(4.36)
with
= 0 ()
k−1
Y
( + ( − j )I );
(4.37)
j=0
R=
k
X
(−1)i
k−1
Y
( + ( − j )I )N i i ( + iI ):
(4.38)
j=i
i=0
It remains to express
Q = Q0 Q1 : : : Qk−1 ;
(4.39)
as Q = SV −1 , with S ∈ B and V a polynomial in . For this we need the following
Lemma 4.5. The operator Q in Eq. (4:39) acting on a vector h = (: : : ; h−1 ; h0 ; h1 ; : : :)T is given by
(Qh)n =
k
X
n
Y
(−1)i+k
i=0
( + + j )
j=n−k+i+1
i (n + + i)
hn+i ;
0 (n + + i + 1)
(4.40)
with i (n) as in Eq. (4:32).
Proof. The proof is by induction on k . For k = 1, we have that
(Q0 h)n = −an
=−
(0)
fn−1
hn + hn+1 ;
fn(0)
(0)
an f˜n−1
(0)
(n + )f˜n
= −(n + + )
using Eq: (3:5);
hn + hn+1 ;
by Eq: (4:14);
0 (n + )
hn + hn+1 ;
0 (n + + 1)
using Eqs: (2:1); (4:16); (4:32) for k = 1:
Suppose now that the result is true for k elementary Darboux transformations, we deduce it for
k + 1 elementary Darboux transformations. Indeed:
(Qk h)n = −an(k)
fn−1
hn + hn+1 ;
fn
= −(n + + − k )
using Eq: (3:5) with f as in Eq: (4:22);
(k) (n + 1) (k+1) (n)
hn + hn+1 ;
(k) (n) (k+1) (n + 1)
by Eqs: (4:19) and (4:23):
(4.41)
F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
291
Assuming Eq. (4.40) with Q as in Eq. (4.39), from the formula (4.41) for Qk , we compute that
the (n; n + i)th entry of QQk , for 06i6k , is given by
(QQk )n; n+i = Qn; n+i−1
− Qn; n+i ( + + n − k + i)
(k) (n + i + 1) (k+1) (n + i)
(k) (n + i) (k+1) (n + i + 1)
(−1)i+k+1 nj=n−k+i ( + + j )
=
(k) (n + i) (k+1) (n + i + 1)
Q
×[i−1 (n + + i − 1) (k+1) (n + i + 1) + i (n + + i) (k+1) (n + i)];
(4.42)
where, in the last equality, we have used Eq. (4.32) and, by convention, we put −1 (n) = 0. Thus, to
establish that Eq. (4.40) holds for QQk , one just needs to show that the term between the brackets
[ : : : ] in the last equality of Eq. (4.42) is
[ : : : ] = (k) (n + i)i(k+1) (n + + i);
(4.43)
where i(k+1) (n) denotes the (k + 1) × (k + 1) determinant obtained by replacing k by k + 1 in the
de nition of i (n), see Eqs. (4.32), (4.16) – (4.18).
We now establish Eq. (4.43) for 16i6k . The case i = 0 is trivial remembering that −1 (n + −
1) = 0. Consider the (k + 2) × (k + 1) matrix
(l−1)
(f˜n+i−k−2+j ) 16j6k+2 ;
16l6k+1
with j and l denoting respectively the line and the column indices. We take V (resp. W ) to be the
subspace of Rk+2 generated by the rst k columns (resp. all the columns) of this matrix. Applying
Lemma 4.4 to this choice of V and W , if we pick in (4.21) (i; j; l) ≡ (1; k + 2 − i; k + 2), we get
that
p2::: k+1 1::: k+2−i:::
1:::k+1 :
[
[
[
k+2 = p1::: k+2−i:::
k+1 2::: k+2 + p2:::k+2−i:::k+2
(4.44)
Using the de nitions (4.16) – (4.18) and (4.32) of the various determinants involved in Eq. (4.43),
one checks easily that this relation amounts precisely to Eq. (4.44), i.e. it is identically satis ed.
This concludes the proof of Lemma 4.5.
It follows immediately from Lemma 4.5, that, in terms of matrices, the operator Q takes the form
Q = SV −1 ;
(4.45)
with
S=
k
X
i+k
(−1)
i=0
V = 0 ( + I ):
T
i
k−1
Y
( + ( − j )I )i ();
(4.46)
j=i
(4.47)
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F.A. Grunbaum et al. / Journal of Computational and Applied Mathematics 106 (1999) 271–297
By combining the results of this section we obtain
Theorem 3. Let be a positive integer and be arbitrary. Then; the (bi-in nite) tridiagonal
matrix which is obtained after k elementary Darboux transformations starting from L( ; ) as in
Eq. (4:1); with 16k6 ; is bispectral.
Proof. The result follows immediately by applying Theorem 1 to the factorization (4.10), with
; R; S and V as in Eqs. (4.37), (4.38), (4.46) and (4.47), using that = b(L( ; )k ) = z k . Given
any family of functions {fn (z )}n ∈ Z belonging to the two-dimensional space of bispectral functions
of L( ; ), which satisfy Eqs. (2.2) and (2.4), we get that the new functions
f˜n = (Pf)n ;
(4.48)
with P as in Eq. (4.33), satisfy
B(k) f˜n = n(k) f˜n ;
(4.49)
with
B(k) = b(R)b(S )z −k ;
n(k) = 0 (n + + 1)0 (n + )
k−1
Y
(n + + − j );
(4.50)
j=0
0 (n) as in Eq. (4.32) and b the bispectral anti-isomorphism de ned in Eqs. (2.16), (2.17), (2:17′ ).
This establishes Theorem 3.
The functions f˜n (z ) that we obtain in Eq. (4.48) by taking f = {fn (z )}n ∈ Z to be the Laguerre
functions introduced in Eq. (2.13), will be called the Krall–Laguerre functions. We remind the
reader (see Remark 4:2) that the matrix P in Eq. (4.48), involved in the Darboux factorization
(4.10), depends on k -free parameters
0 ; : : : ;
k−1 (equivalent to 0 ; : : : ; k−1 in Eq. (4.31)), when
we normalize the functions f( j) in Eq. (4.5) as in Eq. (4.13). Thus, the Krall–Laguerre functions
f˜n (z ) depend on the parameters ( ; ; 0 ; : : : ; k−1 ). When = 0