Differential equations for symmetric generalized ultraspherical polynomials

Author:
Roelof Koekoek

Journal:
Trans. Amer. Math. Soc. **345** (1994), 47-72

MSC:
Primary 33C45; Secondary 34B24, 34L10

DOI:
https://doi.org/10.1090/S0002-9947-1994-1260202-3

MathSciNet review:
1260202

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval with respect to the weight function

In the special case that and we find all differential equations of the form

*n*.

We show that if only for nonnegative integer values of there exists exactly one differential equation which is of finite order .

By using quadratic transformations we also obtain differential equations for the polynomials for all and .

**[1]**T. J. I'a. Bromwich,*An introduction to the theory of infinite series*, 2nd ed., Macmillan, New York, 1959.**[2]**T. S. Chihara,*An introduction to orthogonal polynomials*, Math. and Its Appl., vol. 13, Gordon and Breach, New York, 1978. MR**0481884 (58:1979)****[3]**A. Erdélyi et al. (Eds.),*Higher transcendental functions*, Bateman Manuscript Project, Vol. I, McGraw-Hill, New York, 1953.**[4]**W. N. Everitt and L. L. Littlejohn,*Orthogonal polynomials and spectral theory*:*a survey*, Orthogonal Polynomials and their Applications (C Brezinski, L. Gori, and A. Ronveaux, eds.), IMACS Annals on Computing and Applied Mathematics, vol. 9, J. C. Baltzer A. G., 1991, pp. 21-55. MR**1270216 (95j:34121)****[5]**J. Koekoek and R. Koekoek,*On a differential equation for Koornwinder's generalized Laguerre polynomials*, Proc. Amer. Math. Soc.**112**(1991), 1045-1054. MR**1047003 (91j:33008)****[6]**R. Koekoek,*The search for differential equations for certain sets of orthogonal polynomials*, J. Comput. Appl. Math.**49**(1993), 111-119. MR**1256017 (95m:33008)****[7]**T. H. Koornwinder,*Orthogonal polynomials with weight function*, Canad. Math. Bull. (2)**27**(1984), 205-214. MR**740416 (85i:33011)****[8]**A. M. Krall,*Orthogonal polynomials satisfying fourth order differential equations*, Proc. Royal Soc. Edinburgh Sect. A**87**(1981), 271-288. MR**606336 (82d:33021)****[9]**A. M. Krall and L. L. Littlejohn,*On the classification of differential equations having orthogonal polynomial solutions*. II, Ann. Mat. Pura Appl. (4)**149**(1987), 77-102. MR**932778 (89e:34017)****[10]**H. L. Krall,*Certain differential equations for Tchebycheff polynomials*, Duke Math. J.**4**(1938), 705-718. MR**1546091****[11]**-,*On orthogonal polynomials satisfying a certain fourth order differential equation*, The Pennsylvania State College Studies, No. 6, 1940. MR**0002679 (2:98a)****[12]**L. L. Littlejohn,*The Krall polynomials*:*A new class of orthogonal polynomials*, Quaestiones Math.**5**(1982), 255-265. MR**690030 (84c:42036)****[13]**-,*The Krall polynomials as solutions to a second order differential equation*, Canad. Math. Bull.**26**(1983), 410-417. MR**716580 (85c:33007)****[14]**-,*On the classification of differential equations having orthogonal polynomial solutions*, Ann. Mat. Pura Appl. (4)**93**(1984), 35-53. MR**779537 (86c:33018)****[15]**L. L. Littlejohn and S. D. Shore,*Nonclassical orthogonal polynomials as solutions to second order differential equations*, Canad. Math. Bull.**25**(1982), 291-295. MR**668944 (83i:33006)****[16]**G. Szegö,*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1939; 4th ed., 1975.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
33C45,
34B24,
34L10

Retrieve articles in all journals with MSC: 33C45, 34B24, 34L10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1260202-3

Article copyright:
© Copyright 1994
American Mathematical Society