Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:
Journal of Computational and Applied Mathematics 99 (1998) 401–412
Some properties of the Sobolev–Laguerre polynomials
Jet Wimp ∗ , Harry Kiesel
Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA
Received 5 November 1997; received in revised form 8 June 1998
Abstract
In this paper, we study the case = 0 of the Sobolev–Laguerre polynomials. We determine a generating function for
c 1998 Elsevier Science B.V. All rights reserved.
the polynomials and an expansion formula
AMS classi cation: 33C49; 33C99
Keywords: Sobolev orthogonal polynomials; Laguerre polynomials; Sobolev–Laguerre polynomials; Bessel polynomials;
Reproducing kernels; Generating functions; Laplace transforms
1. Introduction
The so-called Sobolev–Laguerre polynomials have been discussed by a number of di erent writers.
The (general) Sobolev–Laguerre polynomials are orthogonal with respect to the inner product
∗
( )hf; gi =
Z
0
∞
−x
x e f(x)g(x) dx +
Z
∞
x e−x f′ (x)g′ (x) dx;
¿ 0:
0
The polynomials were exhaustively studied by Brenner in the case = 0 who found a (not very
useful) explicit formula for the polynomials [1]. For the more recently investigated case of general
, the article by Marcellan et al. [4] provides an excellent and timely bibliography, as well as
a number of original ndings.
In this paper, we study the case = 0. The notation we use for these polynomials is Ln (x). We
determine an elegant generating function for the polynomials and some expansions involving the
polynomials. These results seem to be new.
∗
Corresponding author. E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 7 3 - 3
402
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
2. Results
We will employ the notation: if A is a matrix with elements ai; j , then A will denote the determinant
whose elements are ai; j , and vice versa.
In our determinants and matrices, i will denote row position, j column position.
We can write Ln (x) as an (n + 1) × (n + 1) Gram-type determinant of moments and powers of x,
i=0::: n
Ln (x) = (ci; j ) j=0:::
n−1
where
i
j
ci; j = hx ; x i =
(
1
x
x 2 ;
..
.
xn
(1)
(i + j)!
i = 0 or j = 0;
(i + j)! + i j(i + j − 2)! otherwise:
(2)
(It is easily veri ed that (1) provides a legitimate expression for Ln (x) by taking the inner product
of the determinant with xi ; i = 0; 1; : : : ; n − 1. The last column of the resulting determinant is equal
to another column, so the result is 0.)
With these polynomials are associated two normalization constants, kn() , the coecient of the
highest power of x in Ln() , and
hn() = hLn() ; Ln() i:
(3)
Taking inner products of the determinant (1) with Ln() shows
()
;
hn() = kn() kn+1
(4)
a useful formula.
It is convenient to factor (i − 1)! out of the ith row, i = 1; 2; : : : ; n + 1, and (j − 1)! out of the
jth column, j = 1; 2; : : : ; n of Ln() and to consider instead the polynomial
i=0::: n
Vn = (di; j ) j=0:::
n−1
where
di; j =
(
1
1 1
1
1
x 1 + 2 + 3
x 2 =2 = 1 + 3 2 + 6
..
..
.. ..
.
.
. .
n
1
:
:
x =n!
:::
:::
:::
..
.
:
i = 0 or j = 0;
[(i + j)! + i j(i + j − 2)!]=i!j! otherwise;
= n L() ;
n
x n =n!
1
x
2
x =2
..
.
(5)
(6)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
403
and
n =
n
Y
j=1
−1
j!(j − 1)! :
(7)
We will investigate three (n + 1) × (n + 1) determinants, which are auxiliary polynomials in x of
degree n:
i=0::: n
Pn (x) = (di; j ) j=0:::
n−1
(d ) i=1::: n+1
Qn (x) = i; j j=1:::
n
i=0::: n
Rn (x) = (di; j ) j=1:::
n
1 1
1
1
: : : 1
x 1 + 2 + 3 : : : x
x 2 = 1 + 3 2 + 6 : : : x 2 ;
..
..
..
..
.. ..
.
.
.
.
. .
n
n
1
:
:
:
x
x
(8)
1 + 2
+3
+ 4 : : : 1
x + 3 2 + 6 3 + 10 : : : x
x 2 + 4 3 + 10 6 + 20 : : : x 2
;
=
..
..
..
..
.. ...
.
.
.
.
.
n
:
:
:
: xn
x
1 1
1
1
x + 2 + 3
+4
x 2 = + 3 2 + 6 3 + 10
..
..
.. ..
.
.
. .
n
:
:
:
x
(9)
: : : 1
: : : x
: : : x2 :
..
..
.
.
: xn
(10)
Note that Ln (x) is essentially the inverse Laplace transform of (1=p) Pn (1=p). This crucial fact
will be used later.
We will also investigate the three following n × n constant (independent of x) determinants:
1
1
(di; j ) i=0::: n−1
An =
=
1
j=0::: n−1
.
..
+ 2
+ 3
(di; j ) i=1::: n
Bn =
=
+ 4
j=1::: n
.
..
1
1
: : :
+ 2 + 3 : : :
+ 3 2 + 6 : : : ;
..
..
..
.
.
.
(11)
+3
+4
: : :
2 + 6 3 + 10 : : :
3 + 10 6 + 20 : : : ;
..
..
..
.
.
.
(12)
404
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
1
+2
(di; j ) i=0::: n−1
Cn =
=
+3
j=1::: n
.
..
1
1
+3
+4
2 + 6 3 + 10
..
..
.
.
: : :
: : :
: : : :
..
.
(13)
Our proofs of results pertaining to these quantities will utilize the three n × n elementary matrices
1
0 0
−1
1
0
0 −1 1
En =
0
0
1 −1
1
0
0
0
Gn =
0
0
0
0 0
0 0
0 0
:::
:::
:::
..
.
:::
0
−1
1
:::
:::
:::
..
.
0
:::
;
0
Fn = EnT ;
1
−1 1
(14)
0 0
0 0
0 0
1
0
:
0
1
Note that Gn di ers from Fn by a single element, having 0 in the (n − 1; n) position, rather than −1.
Note En = Fn = Gn = 1.
Theorem 1. Let
= 4 sinh2 :
(15)
Then
cosh(2n − 1)
;
cosh
sinh(2n + 2)
;
Bn =
sinh 2
Cn = 1:
(16)
An =
(17)
(18)
Proof. A simple computation shows that
1 +1
1
1
0
1
+
2
+
3
0
1
+
3
2
+
6
E n C n Fn =
0
1
+
4
3
+
10
..
..
..
..
.
.
.
.
T
:::
:::
:::
:
:::
..
.
Taking determinants gives Cn = Cn−1 and induction immediately gives the third statement.
(19)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
405
Next, we have
1
0
0
0
0 +1
1
1
1
+2 +3
E n A n Fn = 0
0
1
+ 3 2 + 6
..
..
..
..
.
.
.
.
:::
:::
:::
:
:::
..
.
(20)
Taking determinants, expanding the determinant on the right by minors of the rst column, and then
expanding that determinant by minors of its rst column give
An = An−1 + Bn−2 :
(21)
Applying Sylvester’s theorem (see [7, p. 251]) to An and utilizing the known value of Cn give
An Bn−2 = An−1 Bn−1 − 1:
(22)
Solving this equation for Bn−2 and substituting it into the previous equation yields
A2n = An+1 An−1 − :
(23)
Induction on this equation gives the An result.
Finally, substituting the known value of An into (21) gives the Bn result.
Theorem 2. Let
D = x2 − (1 − x):
Then
(i)
Pn =
(−1)n (x − 1) x(x − 1)n
+
[(x − 1)An + An+1 ];
D
D
Qn =
(−1)n (x − 1)n+1
+
[( + x)An+2 − xAn+1 ];
D
D
(ii)
(iii)
n
Rn = (−1)
"
#
An+1 − (1 − x)An+2 − x(1 − x)n+1
:
D
Proof. Considering the determinants of the matrices En+1 Pn Gn+1 and En+1 Rn Gn+1 in much the same
way as in the proof of Theorem 1 shows that
Pn = (x − 1)Pn−1 + x(x − 1)Qn−2 ;
(24)
Rn = (x − 1)Rn−1 + (−1)n An+1 :
(25)
406
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Applying Sylvester’s theorem to Pn gives
Pn Bn−1 = An xQn−1 − Rn−1 :
(26)
We write (25) as
Rn
Rn−1
(−1)n cosh(2n + 1)
−
=
:
(x − 1)n (x − 1)n−1
cosh (x − 1)n
(27)
Expressing the hyperbolic functions as exponentials and iterating this expression, i.e., replacing n by
k and summing from k = 1 to n (note (25) requires interpreting R0 = 1), give, after some numbing
algebra, (iii).
We now solve (24) for Qn−2 , replace n by n + 1, substitute the result into (26), substitute the
just obtained value of Rn into (26), and nally solve for Bn−1 from (21) and substitute that into the
equation. We obtain the following rst-order di erence equation for Pn :
Pn An+1
Pn+1 An
An+1
An
x
−
=
+ (−1)n
(−1)n
− :
n
n+1
n
n−1
(x − 1)
(x − 1)
D
(x − 1)
(x − 1)
D
(28)
It is easily veri ed, using the known properties of An , that particular solutions corresponding to the
rst and to the second term on the right are, respectively,
(−1)n (x − 1)
;
D
A homogeneous solution is
Pn(2) =
Pn(1) =
An+1 x(x − 1)n
:
D
(29)
Pn(3) = An (x − 1)n :
(30)
We write
Pn = Pn(1) + Pn(2) + CPn(3) :
(31)
C is determined by setting n = 1 and using P1 = x − 1. The result is (i). Finally, using the above
value of Pn we solve for Qn from (24). The result is then simpli ed by expressing An+3 in terms of
An+2 and An+1 and the result is (ii).
In what follows, we let
a=
√
− − 2 +
2
2
+ 4
;
n =
n
Y
j=1
−1
j!(j − 1)! :
(32)
Theorem 3. For |t|¡ min{1; |a|=|x − 1|};
h(x; t) =
∞
X
Pn (x)(−t)n ;
(33)
n=0
where
h(x; t) =
[(1 − t)2 + t(x − 1) + tx(1 − t)]
;
(1 − t)U
(34)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
407
and
U = (t(x − 1) + 1)2 + t(x − 1):
(35)
Proof. Easy algebra gives
∞
X
n=0
An (−t)n (x − 1)n
∞
X
=
1 + t(x − 1)( + 1)
;
U
1 + t(x − 1)
An+1 (−t) (x − 1) =
:
U
n=0
n
(36)
n
(Note A0 = 1:) Multiplying Theorem 2(i) by (−t)n and summing gives the result. (It requires a rather
formidable amount of algebra to show the factor D cancels out of the right-hand side.) The region
of convergence of the series is easily determined by examining the location of the poles of the
left-hand side. These occur at t = 1; t = a=(x − 1); t = 1=[a(x − 1)]. Since |a|¡1¡|1=a|, the region
of convergence is as indicated.
Theorem 4. For |t|¡|a|;
∞
ae xt=(t+a) − e xt=(t+1=a) X
n Ln() (x)(−t)n :
=
(a − 1)(1 − t)
n=0
(37)
Proof. We observe that
1
1
Pn
p
p
= n L{Ln() (x)};
(38)
where L is the Laplace transform operator,
L{f(x)} =
Z
∞
e −px f(x) dx:
(39)
0
Replacing x by 1=p in (33) gives
∞
X
1
n=0
p
Pn
p(1 − t)2 + t(1 − p) + t(1 − t)
1
(−t)n =
:
p
(1 − t)(p2 + (2 + )tp(1 − p) + t 2 (1 − p)2 )
The right-hand side as a function of p has two poles which are; generally, simple.
!
√
t 2t − − 2 ± 4 + 2
±
p =
:
2
(1 − t)2 − t
(40)
(41)
Multiplying by the factor e px and inverting the Laplace transform by integrating along the usual
complex contour gives the result. By an examination of the poles of the denominator as a function
of t; we see that the series (40) converges for all |t|¡|a||p=(p−1)| and thus the complex integration
along any p-contour of the type c−i∞ to c+i∞; c¿1; is justi ed. The result is an analytic function
of t. Since |a|¡1¡|1=a| the series (37) converges for |t|¡|a|.
408
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Note when = 0; (37) yields the standard generating function for the Laguerre polynomials,
[3, vol. 2, p. 189, formula (21)]. To see this, observe that
(−1)n n Ln() (x)|x=0 = Cn = 1:
(42)
The Laguerre polynomials are normalized to have the value 1 at x = 0. Thus,
(−1)n n L(0)
n (x) = Ln (x):
(43)
An interesting inequality for Ln() follows from the generating function (37). The singularities of the
function on the left are at t = 1; t = −a; t = −1=a. Expressed in terms of these are t = 1; t = e −2 ;
t = e 2 . The smallest of these is e −2 . Applying Cauchy’s inequality shows that for any ; 0¡¡e −2
|n Ln() | 6
M
;
− )n
(44)
(e −2
where M is some constant independent of x.
Theorem 5.
e −cx =
∞
X
n Ln() (x);
(45)
n=0
where
n = n (1 − a)
(
c
c+1
n
an−1
+
2n−1
(a
− 1)
c
c+1
n+1
an
2n+1
(a
− 1)
)
;
(46)
and
√
− − 2 + 2 + 4
a=
;
2
n =
n
Y
j=1
−1
j!(j − 1)! :
(47)
The series converges uniformly on compact subsets of the complex x-plane for all ¿0; c¿0; as
can be seen from the estimate (44).
Proof. Denote the generating on the left of (37) by g(x; t). We have
he −cx ; g(x; t)i =
∞
X
n=0
=−
n he −cx ; Ln() (x)i(−t)n
a(ac + a + 1)
(ac + a + c)
+
:
(a − 1)(ct + ac + a) (a − 1)(cat + c + 1)
(48)
409
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Selecting the coecient of t n from the last expression above gives the value of he −cx ; Ln() (x)i. We
now need the value of hn() . However, this is easy to obtain. We have
hg(x; t); g(x; u)i =
∞
X
n=0
=−
2n hLn() (x); Ln() (x)i(ut)n =
∞
X
2n hn() (ut)n
n=0
a2 + 1
a
a
+
−
:
2
2
2
2
(a − 1) (1 − ut=a ) (a − 1) (1 − ut) (a − 1) (1 − a2 ut)
(49)
The last expression above is obtained by the evaluation of a Laplace transform integral and partial
fractions. Selecting the coecient of (ut)n from this expression gives hn() ,
hn() =
−(a2n−1 − 1)(a2n+1 − 1)
:
a2n−1 (a − 1)2 n2
(50)
We then divide he−cx ; Ln() (x)i by this expression and do a little algebra to get the coecients given
in formula (46).
Many other expansions of special functions in the polynomials Ln() can be obtained by similarly
manipulating the generating function (37), including expansions for the con
uent hypergeometric
functions and analogous to those expansions in Laguerre polynomials given in [3, vol. 2,
p. 215].
Marcellan et al. have shown that the Laguerre–Sobolev polynomials for general satisfy a thirdorder (four-term) recurrence relation. One starts with the equation [4] (3.6), which relates the general
Sobolev–Laguerre polynomials Qn( ) (x) (orthogonal with respect to the inner product (∗)) and the
monic Laguerre polynomials Lˆ n( ) (x). In the notation of [4],
( )
( )
(x) = Qn( ) (x) + dn−1 ()Qn−1
(x):
Lˆ n( ) (x) + nLˆ n−1
(51)
dn () is a ratio of two Pollaczek polynomials. Now replace n by n − 1 then by n − 2 and each
( )
( )
time use the recurrence for Lˆ n( ) (x) to express Lˆ n−2
(x) in terms of Lˆ n( ) (x) and Lˆ n−1
(x). Eliminating
the latter two quantities from the set of three equations yields the four-term recursion relation given
in [4, p. 258]:
Qn( ) (x)
qn−2 n(n + − 1) ( )
Qn−1 (x)
+ (2n + − 2 − x) +
qn−1
qn−3 (2n + − 2 − x) ( )
+ (n − 1)(n + − 2) 1 +
Qn−2 (x)
qn−2
+
qn−4 (n − 1)(n − 2)
( )
(n + − 2)(n + − 3)Qn−3
(x) = 0:
qn−3
(52)
Using the recurrence in Proposition 3.4 of [4], one nds that qn () may be expressed a linear
combination of Pollaczek polynomials
qn () = ( + 1)n Pn(1− =2)
1
(1− =2)
+ 1; − =2; =2; −
Pn−1
+ 1; − =2; =2; + 1
2
( + 1)
2
;
(53)
410
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
see also [2, p. 185; 3, vol. 2, p. 220]. There are at least two values of for which the above
expression simpli es. When = 0 the explicit value of the Pollaczek polynomials above can be
found from the formulas [5, (12); 6, (4.4)]. The former provides a formula
3 F2
−n; a;
a + 1; + 1
; 1 =
1
1
n!a
−
;
( − a) (a)n+1 ()n+1
(54)
that can be used in the explicit formula for the Pollaczek polynomial given in [6, (4.4)]. We nd
that for the special case b = −a,
Pn() (1; a; −a; c) =
(c + 2 − 1)n+1
c
−1 :
(2 − 1)
(c)n+1
(55)
Using this formula in (53) shows
qn (0) = ( + 1)n ;
(56)
and for this case (52) is easily found to be an iteration of the recurrence of the classical monic
Laguerre polynomial.
A simpli cation of qn () also occurs in the special case = 0, since
Pn(1)
+ 1; 0; 0; c
2
= Un
+1 ;
2
(57)
where Un denotes the Chebyshev polynomial of the second kind. As is well known, these polynomials
are elementary functions,
√
n+1
− a−n−1
− − 2 + 2 + 4
na
:
(58)
Un
; a=
+ 1 = (−1)
2
a − a−1
2
We nd that when = 0,
qn () = n!(−1)
na
n+1
− a−n
;
a−1
√
− − 2 + 2 + 4
a=
:
2
(59)
Eq. (15) shows
a = −e−2 ;
(60)
so
qn () = n!An+1 ;
(61)
p
cosh(2n − 1) ( 1 + =4 +
An =
=
cosh
√
p
=2)2n−1 + ( 1 + =4 −
p
2 1 + =4
√
=2)2n−1
:
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
411
The resulting four-term recurrence relation for the monic ordinary Laguerre–Sobolev polynomials is
quite simple:
nAn−1
(0)
+ (2n − 2 − x) +
Qn−1
(x)
An
(2n − 2 − x)An−2
(0)
+ (n − 1) (n − 2) +
Qn−2
(x)
An−1
Qn(0) (x)
+
(n − 1)(n − 2)2 An−3 (0)
Qn−3 (x) = 0:
An−2
(62)
A referee has pointed out that this recurrence can be obtained directly from the generating function (37) without recourse to Pollaczek polynomials. Rewriting that equation, we have
(1 − t)
∞
X
n Ln() (x)(−t) n
n=0
xt=(t+a)
=
ae
ext=(t+1=a)
−
(a − 1)
(a − 1)
∞
∞
X
a
1
t X
t n
(1 + at)
=
−
Ln (x)(−at) n :
1+
Ln (x) −
a−1
a n=0
a
a−1
n=0
(63)
Comparing coecients of (−t)n gives
n Ln() (x)
+
()
n−1 Ln−1
(x) =
Now,
1
a−1
1
a n−1
−a
n
1
a−1
= (−1) n An ;
1
a n−1
−a
n
(Ln (x) − Ln−1 (x)):
n Ln (x) =
An (0)
Q (x):
n! n
(64)
(65)
When (65) is used and the Laguerre polynomials are eliminated from the relation (64), one recovers
the recurrence (62).
Acknowledgements
The authors are greatly indebted to the referees, whose painstaking and attentive readings allowed
us to correct many misprints. In addition, they showed us how to correct one false result and how
to substantially simplify the recurrence (62).
References
[1] J. Brenner, Uber
eine Erweiterung des Orthogonalitatbegri es bei Polynomen, Proc. Conf. Constructive Theory of
Functions, Akademiai Kiado, Budapest, 1972.
[2] T.S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
412
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
[3] A. Erdelyi et al., Higher Transcendental Functions, vols. 1 and 2, McGraw-Hill, New York, 1953.
[4] F. Marcellan, T.E. Perez, M.A. Pi˜nar, Laguerre–Sobolev orthogonal polynomials, J. Comp. Appl. Math. 71 (1996)
249–265.
[5] J. Wimp, B. Beckermann, Families of some two-point Pade approximants and some 4 F3 (1) identities, SIAM J. Math.
Anal. 26 (1995) 765–773.
[6] J. Wimp, Pollaczek polynomials and Pade approximants: some closed form expressions, J. Comp. Appl. Math. 32
(1990) 301–310.
[7] J. Wimp, Sequence Transformations and Their Applications, Academic Press, New York, 1981.
Some properties of the Sobolev–Laguerre polynomials
Jet Wimp ∗ , Harry Kiesel
Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA
Received 5 November 1997; received in revised form 8 June 1998
Abstract
In this paper, we study the case = 0 of the Sobolev–Laguerre polynomials. We determine a generating function for
c 1998 Elsevier Science B.V. All rights reserved.
the polynomials and an expansion formula
AMS classi cation: 33C49; 33C99
Keywords: Sobolev orthogonal polynomials; Laguerre polynomials; Sobolev–Laguerre polynomials; Bessel polynomials;
Reproducing kernels; Generating functions; Laplace transforms
1. Introduction
The so-called Sobolev–Laguerre polynomials have been discussed by a number of di erent writers.
The (general) Sobolev–Laguerre polynomials are orthogonal with respect to the inner product
∗
( )hf; gi =
Z
0
∞
−x
x e f(x)g(x) dx +
Z
∞
x e−x f′ (x)g′ (x) dx;
¿ 0:
0
The polynomials were exhaustively studied by Brenner in the case = 0 who found a (not very
useful) explicit formula for the polynomials [1]. For the more recently investigated case of general
, the article by Marcellan et al. [4] provides an excellent and timely bibliography, as well as
a number of original ndings.
In this paper, we study the case = 0. The notation we use for these polynomials is Ln (x). We
determine an elegant generating function for the polynomials and some expansions involving the
polynomials. These results seem to be new.
∗
Corresponding author. E-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 7 3 - 3
402
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
2. Results
We will employ the notation: if A is a matrix with elements ai; j , then A will denote the determinant
whose elements are ai; j , and vice versa.
In our determinants and matrices, i will denote row position, j column position.
We can write Ln (x) as an (n + 1) × (n + 1) Gram-type determinant of moments and powers of x,
i=0::: n
Ln (x) = (ci; j ) j=0:::
n−1
where
i
j
ci; j = hx ; x i =
(
1
x
x 2 ;
..
.
xn
(1)
(i + j)!
i = 0 or j = 0;
(i + j)! + i j(i + j − 2)! otherwise:
(2)
(It is easily veri ed that (1) provides a legitimate expression for Ln (x) by taking the inner product
of the determinant with xi ; i = 0; 1; : : : ; n − 1. The last column of the resulting determinant is equal
to another column, so the result is 0.)
With these polynomials are associated two normalization constants, kn() , the coecient of the
highest power of x in Ln() , and
hn() = hLn() ; Ln() i:
(3)
Taking inner products of the determinant (1) with Ln() shows
()
;
hn() = kn() kn+1
(4)
a useful formula.
It is convenient to factor (i − 1)! out of the ith row, i = 1; 2; : : : ; n + 1, and (j − 1)! out of the
jth column, j = 1; 2; : : : ; n of Ln() and to consider instead the polynomial
i=0::: n
Vn = (di; j ) j=0:::
n−1
where
di; j =
(
1
1 1
1
1
x 1 + 2 + 3
x 2 =2 = 1 + 3 2 + 6
..
..
.. ..
.
.
. .
n
1
:
:
x =n!
:::
:::
:::
..
.
:
i = 0 or j = 0;
[(i + j)! + i j(i + j − 2)!]=i!j! otherwise;
= n L() ;
n
x n =n!
1
x
2
x =2
..
.
(5)
(6)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
403
and
n =
n
Y
j=1
−1
j!(j − 1)! :
(7)
We will investigate three (n + 1) × (n + 1) determinants, which are auxiliary polynomials in x of
degree n:
i=0::: n
Pn (x) = (di; j ) j=0:::
n−1
(d ) i=1::: n+1
Qn (x) = i; j j=1:::
n
i=0::: n
Rn (x) = (di; j ) j=1:::
n
1 1
1
1
: : : 1
x 1 + 2 + 3 : : : x
x 2 = 1 + 3 2 + 6 : : : x 2 ;
..
..
..
..
.. ..
.
.
.
.
. .
n
n
1
:
:
:
x
x
(8)
1 + 2
+3
+ 4 : : : 1
x + 3 2 + 6 3 + 10 : : : x
x 2 + 4 3 + 10 6 + 20 : : : x 2
;
=
..
..
..
..
.. ...
.
.
.
.
.
n
:
:
:
: xn
x
1 1
1
1
x + 2 + 3
+4
x 2 = + 3 2 + 6 3 + 10
..
..
.. ..
.
.
. .
n
:
:
:
x
(9)
: : : 1
: : : x
: : : x2 :
..
..
.
.
: xn
(10)
Note that Ln (x) is essentially the inverse Laplace transform of (1=p) Pn (1=p). This crucial fact
will be used later.
We will also investigate the three following n × n constant (independent of x) determinants:
1
1
(di; j ) i=0::: n−1
An =
=
1
j=0::: n−1
.
..
+ 2
+ 3
(di; j ) i=1::: n
Bn =
=
+ 4
j=1::: n
.
..
1
1
: : :
+ 2 + 3 : : :
+ 3 2 + 6 : : : ;
..
..
..
.
.
.
(11)
+3
+4
: : :
2 + 6 3 + 10 : : :
3 + 10 6 + 20 : : : ;
..
..
..
.
.
.
(12)
404
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
1
+2
(di; j ) i=0::: n−1
Cn =
=
+3
j=1::: n
.
..
1
1
+3
+4
2 + 6 3 + 10
..
..
.
.
: : :
: : :
: : : :
..
.
(13)
Our proofs of results pertaining to these quantities will utilize the three n × n elementary matrices
1
0 0
−1
1
0
0 −1 1
En =
0
0
1 −1
1
0
0
0
Gn =
0
0
0
0 0
0 0
0 0
:::
:::
:::
..
.
:::
0
−1
1
:::
:::
:::
..
.
0
:::
;
0
Fn = EnT ;
1
−1 1
(14)
0 0
0 0
0 0
1
0
:
0
1
Note that Gn di ers from Fn by a single element, having 0 in the (n − 1; n) position, rather than −1.
Note En = Fn = Gn = 1.
Theorem 1. Let
= 4 sinh2 :
(15)
Then
cosh(2n − 1)
;
cosh
sinh(2n + 2)
;
Bn =
sinh 2
Cn = 1:
(16)
An =
(17)
(18)
Proof. A simple computation shows that
1 +1
1
1
0
1
+
2
+
3
0
1
+
3
2
+
6
E n C n Fn =
0
1
+
4
3
+
10
..
..
..
..
.
.
.
.
T
:::
:::
:::
:
:::
..
.
Taking determinants gives Cn = Cn−1 and induction immediately gives the third statement.
(19)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
405
Next, we have
1
0
0
0
0 +1
1
1
1
+2 +3
E n A n Fn = 0
0
1
+ 3 2 + 6
..
..
..
..
.
.
.
.
:::
:::
:::
:
:::
..
.
(20)
Taking determinants, expanding the determinant on the right by minors of the rst column, and then
expanding that determinant by minors of its rst column give
An = An−1 + Bn−2 :
(21)
Applying Sylvester’s theorem (see [7, p. 251]) to An and utilizing the known value of Cn give
An Bn−2 = An−1 Bn−1 − 1:
(22)
Solving this equation for Bn−2 and substituting it into the previous equation yields
A2n = An+1 An−1 − :
(23)
Induction on this equation gives the An result.
Finally, substituting the known value of An into (21) gives the Bn result.
Theorem 2. Let
D = x2 − (1 − x):
Then
(i)
Pn =
(−1)n (x − 1) x(x − 1)n
+
[(x − 1)An + An+1 ];
D
D
Qn =
(−1)n (x − 1)n+1
+
[( + x)An+2 − xAn+1 ];
D
D
(ii)
(iii)
n
Rn = (−1)
"
#
An+1 − (1 − x)An+2 − x(1 − x)n+1
:
D
Proof. Considering the determinants of the matrices En+1 Pn Gn+1 and En+1 Rn Gn+1 in much the same
way as in the proof of Theorem 1 shows that
Pn = (x − 1)Pn−1 + x(x − 1)Qn−2 ;
(24)
Rn = (x − 1)Rn−1 + (−1)n An+1 :
(25)
406
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Applying Sylvester’s theorem to Pn gives
Pn Bn−1 = An xQn−1 − Rn−1 :
(26)
We write (25) as
Rn
Rn−1
(−1)n cosh(2n + 1)
−
=
:
(x − 1)n (x − 1)n−1
cosh (x − 1)n
(27)
Expressing the hyperbolic functions as exponentials and iterating this expression, i.e., replacing n by
k and summing from k = 1 to n (note (25) requires interpreting R0 = 1), give, after some numbing
algebra, (iii).
We now solve (24) for Qn−2 , replace n by n + 1, substitute the result into (26), substitute the
just obtained value of Rn into (26), and nally solve for Bn−1 from (21) and substitute that into the
equation. We obtain the following rst-order di erence equation for Pn :
Pn An+1
Pn+1 An
An+1
An
x
−
=
+ (−1)n
(−1)n
− :
n
n+1
n
n−1
(x − 1)
(x − 1)
D
(x − 1)
(x − 1)
D
(28)
It is easily veri ed, using the known properties of An , that particular solutions corresponding to the
rst and to the second term on the right are, respectively,
(−1)n (x − 1)
;
D
A homogeneous solution is
Pn(2) =
Pn(1) =
An+1 x(x − 1)n
:
D
(29)
Pn(3) = An (x − 1)n :
(30)
We write
Pn = Pn(1) + Pn(2) + CPn(3) :
(31)
C is determined by setting n = 1 and using P1 = x − 1. The result is (i). Finally, using the above
value of Pn we solve for Qn from (24). The result is then simpli ed by expressing An+3 in terms of
An+2 and An+1 and the result is (ii).
In what follows, we let
a=
√
− − 2 +
2
2
+ 4
;
n =
n
Y
j=1
−1
j!(j − 1)! :
(32)
Theorem 3. For |t|¡ min{1; |a|=|x − 1|};
h(x; t) =
∞
X
Pn (x)(−t)n ;
(33)
n=0
where
h(x; t) =
[(1 − t)2 + t(x − 1) + tx(1 − t)]
;
(1 − t)U
(34)
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
407
and
U = (t(x − 1) + 1)2 + t(x − 1):
(35)
Proof. Easy algebra gives
∞
X
n=0
An (−t)n (x − 1)n
∞
X
=
1 + t(x − 1)( + 1)
;
U
1 + t(x − 1)
An+1 (−t) (x − 1) =
:
U
n=0
n
(36)
n
(Note A0 = 1:) Multiplying Theorem 2(i) by (−t)n and summing gives the result. (It requires a rather
formidable amount of algebra to show the factor D cancels out of the right-hand side.) The region
of convergence of the series is easily determined by examining the location of the poles of the
left-hand side. These occur at t = 1; t = a=(x − 1); t = 1=[a(x − 1)]. Since |a|¡1¡|1=a|, the region
of convergence is as indicated.
Theorem 4. For |t|¡|a|;
∞
ae xt=(t+a) − e xt=(t+1=a) X
n Ln() (x)(−t)n :
=
(a − 1)(1 − t)
n=0
(37)
Proof. We observe that
1
1
Pn
p
p
= n L{Ln() (x)};
(38)
where L is the Laplace transform operator,
L{f(x)} =
Z
∞
e −px f(x) dx:
(39)
0
Replacing x by 1=p in (33) gives
∞
X
1
n=0
p
Pn
p(1 − t)2 + t(1 − p) + t(1 − t)
1
(−t)n =
:
p
(1 − t)(p2 + (2 + )tp(1 − p) + t 2 (1 − p)2 )
The right-hand side as a function of p has two poles which are; generally, simple.
!
√
t 2t − − 2 ± 4 + 2
±
p =
:
2
(1 − t)2 − t
(40)
(41)
Multiplying by the factor e px and inverting the Laplace transform by integrating along the usual
complex contour gives the result. By an examination of the poles of the denominator as a function
of t; we see that the series (40) converges for all |t|¡|a||p=(p−1)| and thus the complex integration
along any p-contour of the type c−i∞ to c+i∞; c¿1; is justi ed. The result is an analytic function
of t. Since |a|¡1¡|1=a| the series (37) converges for |t|¡|a|.
408
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Note when = 0; (37) yields the standard generating function for the Laguerre polynomials,
[3, vol. 2, p. 189, formula (21)]. To see this, observe that
(−1)n n Ln() (x)|x=0 = Cn = 1:
(42)
The Laguerre polynomials are normalized to have the value 1 at x = 0. Thus,
(−1)n n L(0)
n (x) = Ln (x):
(43)
An interesting inequality for Ln() follows from the generating function (37). The singularities of the
function on the left are at t = 1; t = −a; t = −1=a. Expressed in terms of these are t = 1; t = e −2 ;
t = e 2 . The smallest of these is e −2 . Applying Cauchy’s inequality shows that for any ; 0¡¡e −2
|n Ln() | 6
M
;
− )n
(44)
(e −2
where M is some constant independent of x.
Theorem 5.
e −cx =
∞
X
n Ln() (x);
(45)
n=0
where
n = n (1 − a)
(
c
c+1
n
an−1
+
2n−1
(a
− 1)
c
c+1
n+1
an
2n+1
(a
− 1)
)
;
(46)
and
√
− − 2 + 2 + 4
a=
;
2
n =
n
Y
j=1
−1
j!(j − 1)! :
(47)
The series converges uniformly on compact subsets of the complex x-plane for all ¿0; c¿0; as
can be seen from the estimate (44).
Proof. Denote the generating on the left of (37) by g(x; t). We have
he −cx ; g(x; t)i =
∞
X
n=0
=−
n he −cx ; Ln() (x)i(−t)n
a(ac + a + 1)
(ac + a + c)
+
:
(a − 1)(ct + ac + a) (a − 1)(cat + c + 1)
(48)
409
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
Selecting the coecient of t n from the last expression above gives the value of he −cx ; Ln() (x)i. We
now need the value of hn() . However, this is easy to obtain. We have
hg(x; t); g(x; u)i =
∞
X
n=0
=−
2n hLn() (x); Ln() (x)i(ut)n =
∞
X
2n hn() (ut)n
n=0
a2 + 1
a
a
+
−
:
2
2
2
2
(a − 1) (1 − ut=a ) (a − 1) (1 − ut) (a − 1) (1 − a2 ut)
(49)
The last expression above is obtained by the evaluation of a Laplace transform integral and partial
fractions. Selecting the coecient of (ut)n from this expression gives hn() ,
hn() =
−(a2n−1 − 1)(a2n+1 − 1)
:
a2n−1 (a − 1)2 n2
(50)
We then divide he−cx ; Ln() (x)i by this expression and do a little algebra to get the coecients given
in formula (46).
Many other expansions of special functions in the polynomials Ln() can be obtained by similarly
manipulating the generating function (37), including expansions for the con
uent hypergeometric
functions and analogous to those expansions in Laguerre polynomials given in [3, vol. 2,
p. 215].
Marcellan et al. have shown that the Laguerre–Sobolev polynomials for general satisfy a thirdorder (four-term) recurrence relation. One starts with the equation [4] (3.6), which relates the general
Sobolev–Laguerre polynomials Qn( ) (x) (orthogonal with respect to the inner product (∗)) and the
monic Laguerre polynomials Lˆ n( ) (x). In the notation of [4],
( )
( )
(x) = Qn( ) (x) + dn−1 ()Qn−1
(x):
Lˆ n( ) (x) + nLˆ n−1
(51)
dn () is a ratio of two Pollaczek polynomials. Now replace n by n − 1 then by n − 2 and each
( )
( )
time use the recurrence for Lˆ n( ) (x) to express Lˆ n−2
(x) in terms of Lˆ n( ) (x) and Lˆ n−1
(x). Eliminating
the latter two quantities from the set of three equations yields the four-term recursion relation given
in [4, p. 258]:
Qn( ) (x)
qn−2 n(n + − 1) ( )
Qn−1 (x)
+ (2n + − 2 − x) +
qn−1
qn−3 (2n + − 2 − x) ( )
+ (n − 1)(n + − 2) 1 +
Qn−2 (x)
qn−2
+
qn−4 (n − 1)(n − 2)
( )
(n + − 2)(n + − 3)Qn−3
(x) = 0:
qn−3
(52)
Using the recurrence in Proposition 3.4 of [4], one nds that qn () may be expressed a linear
combination of Pollaczek polynomials
qn () = ( + 1)n Pn(1− =2)
1
(1− =2)
+ 1; − =2; =2; −
Pn−1
+ 1; − =2; =2; + 1
2
( + 1)
2
;
(53)
410
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
see also [2, p. 185; 3, vol. 2, p. 220]. There are at least two values of for which the above
expression simpli es. When = 0 the explicit value of the Pollaczek polynomials above can be
found from the formulas [5, (12); 6, (4.4)]. The former provides a formula
3 F2
−n; a;
a + 1; + 1
; 1 =
1
1
n!a
−
;
( − a) (a)n+1 ()n+1
(54)
that can be used in the explicit formula for the Pollaczek polynomial given in [6, (4.4)]. We nd
that for the special case b = −a,
Pn() (1; a; −a; c) =
(c + 2 − 1)n+1
c
−1 :
(2 − 1)
(c)n+1
(55)
Using this formula in (53) shows
qn (0) = ( + 1)n ;
(56)
and for this case (52) is easily found to be an iteration of the recurrence of the classical monic
Laguerre polynomial.
A simpli cation of qn () also occurs in the special case = 0, since
Pn(1)
+ 1; 0; 0; c
2
= Un
+1 ;
2
(57)
where Un denotes the Chebyshev polynomial of the second kind. As is well known, these polynomials
are elementary functions,
√
n+1
− a−n−1
− − 2 + 2 + 4
na
:
(58)
Un
; a=
+ 1 = (−1)
2
a − a−1
2
We nd that when = 0,
qn () = n!(−1)
na
n+1
− a−n
;
a−1
√
− − 2 + 2 + 4
a=
:
2
(59)
Eq. (15) shows
a = −e−2 ;
(60)
so
qn () = n!An+1 ;
(61)
p
cosh(2n − 1) ( 1 + =4 +
An =
=
cosh
√
p
=2)2n−1 + ( 1 + =4 −
p
2 1 + =4
√
=2)2n−1
:
J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
411
The resulting four-term recurrence relation for the monic ordinary Laguerre–Sobolev polynomials is
quite simple:
nAn−1
(0)
+ (2n − 2 − x) +
Qn−1
(x)
An
(2n − 2 − x)An−2
(0)
+ (n − 1) (n − 2) +
Qn−2
(x)
An−1
Qn(0) (x)
+
(n − 1)(n − 2)2 An−3 (0)
Qn−3 (x) = 0:
An−2
(62)
A referee has pointed out that this recurrence can be obtained directly from the generating function (37) without recourse to Pollaczek polynomials. Rewriting that equation, we have
(1 − t)
∞
X
n Ln() (x)(−t) n
n=0
xt=(t+a)
=
ae
ext=(t+1=a)
−
(a − 1)
(a − 1)
∞
∞
X
a
1
t X
t n
(1 + at)
=
−
Ln (x)(−at) n :
1+
Ln (x) −
a−1
a n=0
a
a−1
n=0
(63)
Comparing coecients of (−t)n gives
n Ln() (x)
+
()
n−1 Ln−1
(x) =
Now,
1
a−1
1
a n−1
−a
n
1
a−1
= (−1) n An ;
1
a n−1
−a
n
(Ln (x) − Ln−1 (x)):
n Ln (x) =
An (0)
Q (x):
n! n
(64)
(65)
When (65) is used and the Laguerre polynomials are eliminated from the relation (64), one recovers
the recurrence (62).
Acknowledgements
The authors are greatly indebted to the referees, whose painstaking and attentive readings allowed
us to correct many misprints. In addition, they showed us how to correct one false result and how
to substantially simplify the recurrence (62).
References
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eine Erweiterung des Orthogonalitatbegri es bei Polynomen, Proc. Conf. Constructive Theory of
Functions, Akademiai Kiado, Budapest, 1972.
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J. Wimp, H. Kiesel / Journal of Computational and Applied Mathematics 99 (1998) 401–412
[3] A. Erdelyi et al., Higher Transcendental Functions, vols. 1 and 2, McGraw-Hill, New York, 1953.
[4] F. Marcellan, T.E. Perez, M.A. Pi˜nar, Laguerre–Sobolev orthogonal polynomials, J. Comp. Appl. Math. 71 (1996)
249–265.
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Anal. 26 (1995) 765–773.
[6] J. Wimp, Pollaczek polynomials and Pade approximants: some closed form expressions, J. Comp. Appl. Math. 32
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