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) 189–194
Variations on a theme of Heine and Stieltjes: an electrostatic
interpretation of the zeros of certain polynomials
F. Alberto Grunbaum 1
Department of Mathematics, University of California, Berkeley, CA 94720, USA
Received 15 January 1998
Abstract
We show a way to adapt the ideas of Stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of
c 1998 Elsevier Science B.V. All rights reserved.
orthogonal polynomials.
Keywords: Orthogonal polynomials; Logarithmic potential; Koornwinder polynomials
1. Introduction
Given a sequence of polynomials pn (x) with deg pn (x) = n one can ask for a physical interpretation
of its roots. A nice instance of this is given by Stieltjes’ observation [21] in connection with earlier
work of Heine [12]. The results of Stieltjes apply to a large class of polynomials, including the
Lame and the Jacobi polynomials. At about the same time Stieltjes published three other notes in
connection with this problem [22–24]. These results were later extended to other classical orthogonal
polynomials, as described in Szego [25]. See also [6]. All these polynomials satisfy second order
di erential equations, and the crucial importance of this fact is apparent in the title of [1].
My interest in this topic arose in connection with the bispectral problem in the discrete–continuous
context, i.e., for orthogonal polynomials in a continuous variable. This is not the appropriate place
to recall this material and I refer the reader to [5, 7–10]. The main point of this line of work is that
one encounters orthogonal polynomials satisfying higher order di erential equations of the type rst
discovered by Krall [15]. If one looks for a ‘physical interpretation’ of the zeros of these families of
1
The author was supported in part by NSF grant # DMS94-00097 and by AFOSR under Contract FDF49620-96-1-0127.
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 5 6 - 3
190
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
polynomials in a straightforward adaptation of the methods of Stieltjes one is faced with the need to
postulate rather ‘unphysical’ interactions among charges: the main point is that it appears neccesary
to consider not just pairwise interactions but more general N -body interactions. While these may be
of some interest in certain cases of high density plasmas, they remain rather hard to motivate.
The main point of this note is to observe that a simple adaptation of the ideas of Heine and
Stieltjes allows one to give an interpretation within the realm of pairwise interactions. We choose
to illustrate the phenomenon in the case of a family of polynomials introduced by Koornwinder
[14], but it is clear that the same analysis can be done for many other such families: all that is
needed is that the polynomials should satisfy second order di erential equations where we allow the
coecients to depend on the degree n of the polynomials, i.e.
(a(n; x)D2 + b(n; x)D + c(n; x))pn (x) = 0:
There are large classes of polynomials that meet these conditions, and we just mention some papers
where this issue has been considered in the context of orthogonal polynomials, namely [1, 11, 17–20].
See also [2–4].
During the preparation of this note I got a preprint by Mourad Ismail [13] where he considers a
very general class of polynomials and develops ideas that are related to those discussed here. He
nds an expression for the energy at equilibrium in terms of the coecients in the recursion relation
satis ed by the corresponding orthogonal polynomials.
All these interpretations require a logarithmic potential in dimension one, and thus one is thinking
of in nitely long parallel wires in dimension three that interact with the usual Coulomb potential.
I will have a word on Coulomb interaction in dimension one at the very end of the paper.
The reader will notice that one does not need to require that pn (x) should be a family of orthogonal
polynomials. A ‘naturally appearing’ class of polynomials where this should work are the Krall–
Hermite and Krall–Bessel polynomials, see [10], as well as some extensions of the more classical
Krall polynomials, see [9].
2. The work of Stieltjes and Heine
The contributions in [12, 21] can be described as follows.
Consider a total of p+1 positive charges that are nailed down at xed locations t1 ¡t2 ¡ · · · ¡tp+1
on the real line. The charge at location ti is given by the positive quantity i , i = 1; 2; : : : ; p + 1.
Consider now n unit positive charges that are distributed in between t1 and tp+1 , i.e. in the p
open intervals (t1 ; t2 ); (t2 ; t3 ); : : : ; (tp ; tp+1 ).
This can be done in a total of
n+p−1
n
ways. These arrangements of charges are determined by the choice of nonnegative integers
n1 ; n2 ; : : : ; np
such that n1 + n2 + np = n.
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
191
Here nk denotes the number of ‘movable charges’ that are placed between the charges held xed
at tk and tk+1 .
We will assume all along that these ‘charges’ interact with other charges through a logarithmic
potential.
It is entirely reasonable that for each such arrangement of n charges there is exactly one ‘equilibrium position’. We denote these points by x1 ; x2 ; : : : ; xn . Both papers [12, 21] contain arguments
to support this assertion.
Heine and Stieltjes consider polynomial solutions of the equation
(A(x)D2 + B(x)D + C(x))P(x) = 0;
D = d=dx:
Here A(x) is a polynomial of degree p + 1, A = (x − t1 ); (x − t2 ); : : : ; (x − tp+1 ). B(x) is a polynomial
of degree p such that for positive j ’s
B=A = 2( 1 =(x − t1 ) + 2 =(x − t2 ) + · · · + p+1 =(x − tp+1 )):
P(x) is sought as a polynomial of degree n, and C(x) is searched for as a polynomial of degree
p − 1 that is allowed to depend on the solution P(x) itself.
Now we come to a crucial property of polynomials with simple zeros. If
P(x) = (x − x1 )(x − x2 ) · · · (x − xn )
then at any root xk of P we have
P ′′ (xk ) = 2P ′ (xk )
X
j6=k
1
:
xk − xj
Using this property we observe that if P(x) solves the di erential equation above it follows that at
each xk we have
X
j6=k
1=(xk − xj ) +
p+1
X
i=1
( i =(xk − ti )) = 0:
But this is exactly the ‘electrostatic equilibrium condition’ described above.
In the case p = 1 (Jacobi) there is one such polynomial for each n, in the case of p = 2 (Lame)
there are n + 1 such polynomials, for p = 3 we have (n + 2)(n + 2)=2 polynomials, etc.
3. The Koornwinder polynomials
For the polynomials introduced by Koornwinder [14] which are orthogonal with respect to the
weight function
(1 − x)a (1 + x)b + t(x + 1) + s(x − 1);
one has
un = pn yn + qn yn′ ;
yn = rn un + sn un′ ;
−16x61
192
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
where yn and un are the Jacobi and the Koornwinder polynomials respectively, and pn ; qn ; rn ; sn are
ratios of polynomials of bounded degree.
Then the result derived in [14] (with a minor misprint corrected in [16]) is that the polynomials
un satisfy an equation as in Section 2 with the relevant ratio being given by
B=A = pn =qn + rn =sn + sn′ =sn
and this ratio is expressible in a nice form. For simplicity we give it here for the case a = b, t = s.
We get
B=A = (2 + a)=(x − 1) + (2 + a)=(x + 1) − 1=(x −
√
zn ) − 1=(x +
√
zn )
with
zn =
((2a + n + 2)t + (n − 2)!n=(2dn ))((2a + n + 1)t + (n − 1)!=(2dn ))
:
(t + (n − 2)!=(2dn ))((2a + n + 1)(2a + n + 2)t + n!=(2dn ))
Here dn = (2a + 3); (2a + 4); : : : ; (2a + n). Note that 0¡zn ¡1.
This means that if for each n we x a pair of positive charges of strength 1 + a=2 at each of the
√
√
points +1 and −1 and a pair of negative charges of strength − 12 at the points zn and − zn and we
place n positive unit charges in between the two negative ones, then the equilibrium con guration
– assuming the usual pairwise logarithmic interaction – is given by the zeros of the Koornwinder
polynomials.
Notice that the only di erence with the Stieltjes construction is that we have to allow for xed
charges whose location depends on n and that not all the xed charges are positive.
When t = 0 we have zn = 1 and this results in charges of strength 21 + 12 a at the endpoints +1; −1.
This is the classical result of Stieltjes. Notice that the limit t = ∞ also gives this classical situation.
Finally notice that as n approaches in nity the value of zn tends to 1 too.
For a general Koornwinder polynomial the relevant ratio takes the form
(2 + a)=(x − 1) + (2 + b)=(x + 1) − 1=(x −
√
an ) − 1=(x +
p
bn )
and the expressions for an and bn are a bit ugly to present here.
4. Inverse square law
It would be interesting to see a relation between zeros of well-known families of polynomials and
a one-dimensional electrostatic problem involving the usual Coulomb inverse square law.
We saw earlier that the reason why a logarithmic potential plays such an important role is directly
connected with a simple property of polynomials with simple roots, namely: at each root of P we
have
X
P ′′ (xk )
1
=
:
2P ′ (xk )
x
−
xj
j6=k k
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
193
One can see that a di erent di erential expression involving P has a de nite ‘Coulomb type’
appearance, namely
1
−
3
P ′′′ (xk ) 3
−
P ′ (xk )
4
If the factor
3
4
P ′′ (xk )
P ′ (xk )
2 !
were replaced by
3
2
=
X
j6=k
1
:
(xk − xj )2
we would be dealing with the usual Schwartzian derivative.
References
[1] F.V. Atkinson, W.N. Everitt, Orthogonal polynomials which satisfy second order di erential equations, in: P. Butzer,
F. Feher (Eds.), E.B. Christo el, the in
uence of his work, on Mathematics and Physical Sciences, Birkhauser, Basel,
1981, pp. 173–181.
[2] W. Bauldry, Estimates of asymmetric Freud polynomials on the real line, J. Approximation Theory 63 (1990)
225–237.
[3] S.S. Bonan, D.S. Clark, Estimates of the Hermite and the Freud polynomials, J. Approximation Theory 63 (1990)
210 –224.
[4] Y. Chen, M.E.H. Ismail, Ladder operators and di erential equations for orthogonal polynomials, J. Phys. A 30 (1997)
7818 –7829.
[5] J.J. Duistermaat, F.A. Grunbaum, Di erential equations in the spectral parameter, Comm. Math. Phys. 103 (1986)
177–240.
[6] P.J. Forrester, J.B. Rogers, Electrostatics and the zeros of the classical orthogonal polynomials, SIAM J. Math. Anal.
17 (1986) 461– 468.
[7] F.A. Grunbaum, L. Haine, Orthogonal polynomials satisfying di erential equations: the role of the Darboux
transformation, in: D. Levi, L. Vinet, P. Winternitz (Eds.), Symmetries and Integrability of Di erences Equations
(Esterel, PQ, 1994), CRM Proc. Lecture Notes, vol. 9, Amer. Math. Soc., Providence, RI, 1996, pp. 143–154.
[8] F.A. Grunbaum, L. Haine, A theorem of Bochner, revisited, in: A.S. Fokas, I.M. Gelfand (Eds.), Algebraic Aspects
of Integrable Systems: In memory of Irene Dorfman, Progr. Nonlinear Di erential Equations Appl. 26, Birkhauser,
Boston, 1997, pp. 143 –172.
[9] F.A. Grunbaum, L. Haine, Bispectral Darboux transformations: an extension of the Krall polynomials, IMRN
(Internat. Math. Res. Notices) (1997) 359 – 392.
[10] F.A. Grunbaum, L. Haine, E. Horozov, On the Krall–Hermite and the Krall–Bessel polynomials, IMRN (Internat.
Math. Res. Notices), No. 19 (1997) 953–966.
[11] W. Hahn, On di erential equations for orthogonal polynomials, Funkcialaj Ekvac. 21 (1978) 1–9.
[12] E. Heine, Comptes rendus de l’Academie des sciences de Berlin, 1864.
[13] M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials, Preprint, November 9, 1997.
[14] T.H. Koornwinder, Orthogonal polynomials with weight function (1 − x) (1 + x) + M(x + 1) + N(x − 1), Can.
Math. Bull. 27 (2) (1984) 205–214.
[15] H.L. Krall, Certain di erential equations for Tchebyche polynomials, Duke Math. J. 4 (1938) 705 –718.
[16] L.L. Littlejohn, An application of a new theorem on orthogonal polynomials and di erential equations, Quaestiones
Math. 10 (1) (1986) 49 – 61.
[17] L.L. Littlejohn, S.D. Shore, Nonclassical orthogonal polynomials as solutions of second order di erential equations,
Can. Math. Bull. 25 (3) (1982).
[18] A. Magnus, On Freud’s equations for exponential weights, J. Approx. Theory 46 (1986) 65–99.
[19] P. Nevai, Geza Freud, orthogonal polynomials and Christo el functions. A case study, J. Approx. Theory 48 (1986)
3–167.
[20] J. Shohat, A di erential equation for orthogonal polynomials, Duke Math. J. 5 (1939) 401– 417.
[21] T.J. Stieltjes, Sur certains polynomes qui veri ent une e quation di erentielle lineaire du second ordre et sur la theorie
des fonctions de Lame, Acta Math. 8 (1885) 321–326; Oeuvres Completes 1, 434 – 439.
194
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
[22] T.J. Stieltjes, Sur quelques theoremes d’algebre, Comptes Rendus de l’Academie des Sciences, Paris 100 (1885)
439 – 440; Oeuvres Completes 1, 440 – 441.
[23] T.J. Stieltjes, Sur les polynomes de Jacobi, Comptes Rendus de l’Academie des Sciences, Paris 100 (1885) 620 – 622;
Oeuvres Completes 1, 442 – 444.
[24] T.J. Stieltjes, Sur les racines de l’equation Xn = 0, Acta Math. 9 (1886) 385 – 400; Oeuvres Completes 2, 73–88.
[25] G. Szego, Orthogonal Polynomials, 4th ed., Amer. Math. Soc., Providence, RI, 1975.
Variations on a theme of Heine and Stieltjes: an electrostatic
interpretation of the zeros of certain polynomials
F. Alberto Grunbaum 1
Department of Mathematics, University of California, Berkeley, CA 94720, USA
Received 15 January 1998
Abstract
We show a way to adapt the ideas of Stieltjes to obtain an electrostatic interpretation of the zeroes of a large class of
c 1998 Elsevier Science B.V. All rights reserved.
orthogonal polynomials.
Keywords: Orthogonal polynomials; Logarithmic potential; Koornwinder polynomials
1. Introduction
Given a sequence of polynomials pn (x) with deg pn (x) = n one can ask for a physical interpretation
of its roots. A nice instance of this is given by Stieltjes’ observation [21] in connection with earlier
work of Heine [12]. The results of Stieltjes apply to a large class of polynomials, including the
Lame and the Jacobi polynomials. At about the same time Stieltjes published three other notes in
connection with this problem [22–24]. These results were later extended to other classical orthogonal
polynomials, as described in Szego [25]. See also [6]. All these polynomials satisfy second order
di erential equations, and the crucial importance of this fact is apparent in the title of [1].
My interest in this topic arose in connection with the bispectral problem in the discrete–continuous
context, i.e., for orthogonal polynomials in a continuous variable. This is not the appropriate place
to recall this material and I refer the reader to [5, 7–10]. The main point of this line of work is that
one encounters orthogonal polynomials satisfying higher order di erential equations of the type rst
discovered by Krall [15]. If one looks for a ‘physical interpretation’ of the zeros of these families of
1
The author was supported in part by NSF grant # DMS94-00097 and by AFOSR under Contract FDF49620-96-1-0127.
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 5 6 - 3
190
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
polynomials in a straightforward adaptation of the methods of Stieltjes one is faced with the need to
postulate rather ‘unphysical’ interactions among charges: the main point is that it appears neccesary
to consider not just pairwise interactions but more general N -body interactions. While these may be
of some interest in certain cases of high density plasmas, they remain rather hard to motivate.
The main point of this note is to observe that a simple adaptation of the ideas of Heine and
Stieltjes allows one to give an interpretation within the realm of pairwise interactions. We choose
to illustrate the phenomenon in the case of a family of polynomials introduced by Koornwinder
[14], but it is clear that the same analysis can be done for many other such families: all that is
needed is that the polynomials should satisfy second order di erential equations where we allow the
coecients to depend on the degree n of the polynomials, i.e.
(a(n; x)D2 + b(n; x)D + c(n; x))pn (x) = 0:
There are large classes of polynomials that meet these conditions, and we just mention some papers
where this issue has been considered in the context of orthogonal polynomials, namely [1, 11, 17–20].
See also [2–4].
During the preparation of this note I got a preprint by Mourad Ismail [13] where he considers a
very general class of polynomials and develops ideas that are related to those discussed here. He
nds an expression for the energy at equilibrium in terms of the coecients in the recursion relation
satis ed by the corresponding orthogonal polynomials.
All these interpretations require a logarithmic potential in dimension one, and thus one is thinking
of in nitely long parallel wires in dimension three that interact with the usual Coulomb potential.
I will have a word on Coulomb interaction in dimension one at the very end of the paper.
The reader will notice that one does not need to require that pn (x) should be a family of orthogonal
polynomials. A ‘naturally appearing’ class of polynomials where this should work are the Krall–
Hermite and Krall–Bessel polynomials, see [10], as well as some extensions of the more classical
Krall polynomials, see [9].
2. The work of Stieltjes and Heine
The contributions in [12, 21] can be described as follows.
Consider a total of p+1 positive charges that are nailed down at xed locations t1 ¡t2 ¡ · · · ¡tp+1
on the real line. The charge at location ti is given by the positive quantity i , i = 1; 2; : : : ; p + 1.
Consider now n unit positive charges that are distributed in between t1 and tp+1 , i.e. in the p
open intervals (t1 ; t2 ); (t2 ; t3 ); : : : ; (tp ; tp+1 ).
This can be done in a total of
n+p−1
n
ways. These arrangements of charges are determined by the choice of nonnegative integers
n1 ; n2 ; : : : ; np
such that n1 + n2 + np = n.
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
191
Here nk denotes the number of ‘movable charges’ that are placed between the charges held xed
at tk and tk+1 .
We will assume all along that these ‘charges’ interact with other charges through a logarithmic
potential.
It is entirely reasonable that for each such arrangement of n charges there is exactly one ‘equilibrium position’. We denote these points by x1 ; x2 ; : : : ; xn . Both papers [12, 21] contain arguments
to support this assertion.
Heine and Stieltjes consider polynomial solutions of the equation
(A(x)D2 + B(x)D + C(x))P(x) = 0;
D = d=dx:
Here A(x) is a polynomial of degree p + 1, A = (x − t1 ); (x − t2 ); : : : ; (x − tp+1 ). B(x) is a polynomial
of degree p such that for positive j ’s
B=A = 2( 1 =(x − t1 ) + 2 =(x − t2 ) + · · · + p+1 =(x − tp+1 )):
P(x) is sought as a polynomial of degree n, and C(x) is searched for as a polynomial of degree
p − 1 that is allowed to depend on the solution P(x) itself.
Now we come to a crucial property of polynomials with simple zeros. If
P(x) = (x − x1 )(x − x2 ) · · · (x − xn )
then at any root xk of P we have
P ′′ (xk ) = 2P ′ (xk )
X
j6=k
1
:
xk − xj
Using this property we observe that if P(x) solves the di erential equation above it follows that at
each xk we have
X
j6=k
1=(xk − xj ) +
p+1
X
i=1
( i =(xk − ti )) = 0:
But this is exactly the ‘electrostatic equilibrium condition’ described above.
In the case p = 1 (Jacobi) there is one such polynomial for each n, in the case of p = 2 (Lame)
there are n + 1 such polynomials, for p = 3 we have (n + 2)(n + 2)=2 polynomials, etc.
3. The Koornwinder polynomials
For the polynomials introduced by Koornwinder [14] which are orthogonal with respect to the
weight function
(1 − x)a (1 + x)b + t(x + 1) + s(x − 1);
one has
un = pn yn + qn yn′ ;
yn = rn un + sn un′ ;
−16x61
192
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
where yn and un are the Jacobi and the Koornwinder polynomials respectively, and pn ; qn ; rn ; sn are
ratios of polynomials of bounded degree.
Then the result derived in [14] (with a minor misprint corrected in [16]) is that the polynomials
un satisfy an equation as in Section 2 with the relevant ratio being given by
B=A = pn =qn + rn =sn + sn′ =sn
and this ratio is expressible in a nice form. For simplicity we give it here for the case a = b, t = s.
We get
B=A = (2 + a)=(x − 1) + (2 + a)=(x + 1) − 1=(x −
√
zn ) − 1=(x +
√
zn )
with
zn =
((2a + n + 2)t + (n − 2)!n=(2dn ))((2a + n + 1)t + (n − 1)!=(2dn ))
:
(t + (n − 2)!=(2dn ))((2a + n + 1)(2a + n + 2)t + n!=(2dn ))
Here dn = (2a + 3); (2a + 4); : : : ; (2a + n). Note that 0¡zn ¡1.
This means that if for each n we x a pair of positive charges of strength 1 + a=2 at each of the
√
√
points +1 and −1 and a pair of negative charges of strength − 12 at the points zn and − zn and we
place n positive unit charges in between the two negative ones, then the equilibrium con guration
– assuming the usual pairwise logarithmic interaction – is given by the zeros of the Koornwinder
polynomials.
Notice that the only di erence with the Stieltjes construction is that we have to allow for xed
charges whose location depends on n and that not all the xed charges are positive.
When t = 0 we have zn = 1 and this results in charges of strength 21 + 12 a at the endpoints +1; −1.
This is the classical result of Stieltjes. Notice that the limit t = ∞ also gives this classical situation.
Finally notice that as n approaches in nity the value of zn tends to 1 too.
For a general Koornwinder polynomial the relevant ratio takes the form
(2 + a)=(x − 1) + (2 + b)=(x + 1) − 1=(x −
√
an ) − 1=(x +
p
bn )
and the expressions for an and bn are a bit ugly to present here.
4. Inverse square law
It would be interesting to see a relation between zeros of well-known families of polynomials and
a one-dimensional electrostatic problem involving the usual Coulomb inverse square law.
We saw earlier that the reason why a logarithmic potential plays such an important role is directly
connected with a simple property of polynomials with simple roots, namely: at each root of P we
have
X
P ′′ (xk )
1
=
:
2P ′ (xk )
x
−
xj
j6=k k
F. Alberto Grunbaum / Journal of Computational and Applied Mathematics 99 (1998) 189–194
193
One can see that a di erent di erential expression involving P has a de nite ‘Coulomb type’
appearance, namely
1
−
3
P ′′′ (xk ) 3
−
P ′ (xk )
4
If the factor
3
4
P ′′ (xk )
P ′ (xk )
2 !
were replaced by
3
2
=
X
j6=k
1
:
(xk − xj )2
we would be dealing with the usual Schwartzian derivative.
References
[1] F.V. Atkinson, W.N. Everitt, Orthogonal polynomials which satisfy second order di erential equations, in: P. Butzer,
F. Feher (Eds.), E.B. Christo el, the in
uence of his work, on Mathematics and Physical Sciences, Birkhauser, Basel,
1981, pp. 173–181.
[2] W. Bauldry, Estimates of asymmetric Freud polynomials on the real line, J. Approximation Theory 63 (1990)
225–237.
[3] S.S. Bonan, D.S. Clark, Estimates of the Hermite and the Freud polynomials, J. Approximation Theory 63 (1990)
210 –224.
[4] Y. Chen, M.E.H. Ismail, Ladder operators and di erential equations for orthogonal polynomials, J. Phys. A 30 (1997)
7818 –7829.
[5] J.J. Duistermaat, F.A. Grunbaum, Di erential equations in the spectral parameter, Comm. Math. Phys. 103 (1986)
177–240.
[6] P.J. Forrester, J.B. Rogers, Electrostatics and the zeros of the classical orthogonal polynomials, SIAM J. Math. Anal.
17 (1986) 461– 468.
[7] F.A. Grunbaum, L. Haine, Orthogonal polynomials satisfying di erential equations: the role of the Darboux
transformation, in: D. Levi, L. Vinet, P. Winternitz (Eds.), Symmetries and Integrability of Di erences Equations
(Esterel, PQ, 1994), CRM Proc. Lecture Notes, vol. 9, Amer. Math. Soc., Providence, RI, 1996, pp. 143–154.
[8] F.A. Grunbaum, L. Haine, A theorem of Bochner, revisited, in: A.S. Fokas, I.M. Gelfand (Eds.), Algebraic Aspects
of Integrable Systems: In memory of Irene Dorfman, Progr. Nonlinear Di erential Equations Appl. 26, Birkhauser,
Boston, 1997, pp. 143 –172.
[9] F.A. Grunbaum, L. Haine, Bispectral Darboux transformations: an extension of the Krall polynomials, IMRN
(Internat. Math. Res. Notices) (1997) 359 – 392.
[10] F.A. Grunbaum, L. Haine, E. Horozov, On the Krall–Hermite and the Krall–Bessel polynomials, IMRN (Internat.
Math. Res. Notices), No. 19 (1997) 953–966.
[11] W. Hahn, On di erential equations for orthogonal polynomials, Funkcialaj Ekvac. 21 (1978) 1–9.
[12] E. Heine, Comptes rendus de l’Academie des sciences de Berlin, 1864.
[13] M.E.H. Ismail, An electrostatic model for zeros of general orthogonal polynomials, Preprint, November 9, 1997.
[14] T.H. Koornwinder, Orthogonal polynomials with weight function (1 − x) (1 + x) + M(x + 1) + N(x − 1), Can.
Math. Bull. 27 (2) (1984) 205–214.
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