Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 $ \{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty $ which are orthogonal on the interval $ [- 1,1]$ with respect to the weight function

$\displaystyle \frac{{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Ga... ...ta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),$

where $ \alpha > - 1$, $ \beta > - 1$, $ M \geq 0$, and $ N \geq 0$.

In the special case that $ \beta = \alpha $ and $ N = M$ we find all differential equations of the form

$\displaystyle \sum\limits_{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),} $

where the coefficients $ \{ {c_i}(x)\} _{i = 1}^\infty $ are independent of the degree n.

We show that if $ M > 0$ only for nonnegative integer values of $ \alpha $ there exists exactly one differential equation which is of finite order $ 2\alpha + 4$.

By using quadratic transformations we also obtain differential equations for the polynomials $ \{ P_n^{\alpha, \pm 1/2,0,N}(x)\} _{n = 0}^\infty $ for all $ \alpha > - 1$ and $ N \geq 0$.


References [Enhancements On Off] (What's this?)

  • [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 $ {(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1)$, 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.

Similar Articles

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

American Mathematical Society