Differential equations for symmetric generalized ultraspherical polynomials
Author:
Roelof Koekoek
Journal:
Trans. Amer. Math. Soc. 345 (1994), 4772
MSC:
Primary 33C45; Secondary 34B24, 34L10
MathSciNet review:
1260202
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Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials which are orthogonal on the interval with respect to the weight function where , , , and . In the special case that and we find all differential equations of the form where the coefficients are independent of the degree 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, Gordon
and Breach Science Publishers, New YorkLondonParis, 1978. Mathematics and
its Applications, Vol. 13. MR 0481884
(58 #1979)
 [3]
A. Erdélyi et al. (Eds.), Higher transcendental functions, Bateman Manuscript Project, Vol. I, McGrawHill, New York, 1953.
 [4]
W.
N. Everitt and L.
L. Littlejohn, Orthogonal polynomials and spectral theory: a
survey, Orthogonal polynomials and their applications (Erice, 1990)
IMACS Ann. Comput. Appl. Math., vol. 9, Baltzer, Basel, 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), no. 4, 1045–1054. MR 1047003
(91j:33008), http://dx.doi.org/10.1090/S00029939199110470039
 [6]
Roelof
Koekoek, The search for differential equations for certain sets of
orthogonal polynomials, Proceedings of the Seventh Spanish Symposium
on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991),
1993, pp. 111–119. MR 1256017
(95m:33008), http://dx.doi.org/10.1016/03770427(93)90141W
 [7]
Tom
H. Koornwinder, Orthogonal polynomials with weight function
(1𝑥)^{𝛼}(1+𝑥)^{𝛽}+𝑀𝛿(𝑥+1)+𝑁𝛿(𝑥1),
Canad. Math. Bull. 27 (1984), no. 2, 205–214.
MR 740416
(85i:33011), http://dx.doi.org/10.4153/CMB19840307
 [8]
Allan
M. Krall, Orthogonal polynomials satisfying fourth order
differential equations, Proc. Roy. Soc. Edinburgh Sect. A
87 (1980/81), no. 34, 271–288. MR 606336
(82d:33021), http://dx.doi.org/10.1017/S0308210500015213
 [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), http://dx.doi.org/10.1007/BF01773927
 [10]
H.
L. Krall, Certain differential equations for Tchebycheff
polynomials, Duke Math. J. 4 (1938), no. 4,
705–718. MR
1546091, http://dx.doi.org/10.1215/S0012709438004624
 [11]
H.
L. Krall, On orthogonal polynomials satisfying a certain fourth
order differential equation, Pennsylvania State College Studies,
1940 (1940), no. 6, 24. MR 0002679
(2,98a)
 [12]
Lance
L. Littlejohn, The Krall polynomials: a new class of orthogonal
polynomials, Quaestiones Math. 5 (1982/83),
no. 3, 255–265. MR 690030
(84c:42036)
 [13]
Lance
L. Littlejohn, The Krall polynomials as solutions to a second order
differential equation, Canad. Math. Bull. 26 (1983),
no. 4, 410–417. MR 716580
(85c:33007), http://dx.doi.org/10.4153/CMB19830689
 [14]
Lance
L. Littlejohn, On the classification of differential equations
having orthogonal polynomial solutions, Ann. Mat. Pura Appl. (4)
138 (1984), 35–53. MR 779537
(86c:33018), http://dx.doi.org/10.1007/BF01762538
 [15]
Lance
L. Littlejohn and Samuel
D. Shore, Nonclassical orthogonal polynomials as solutions to
second order differential equations, Canad. Math. Bull.
25 (1982), no. 3, 291–295. MR 668944
(83i:33006), http://dx.doi.org/10.4153/CMB19820402
 [16]
G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1939; 4th ed., 1975.
 [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, McGrawHill, 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. 2155. 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), 10451054. MR 1047003 (91j:33008)
 [6]
 R. Koekoek, The search for differential equations for certain sets of orthogonal polynomials, J. Comput. Appl. Math. 49 (1993), 111119. MR 1256017 (95m:33008)
 [7]
 T. H. Koornwinder, Orthogonal polynomials with weight function , Canad. Math. Bull. (2) 27 (1984), 205214. MR 740416 (85i:33011)
 [8]
 A. M. Krall, Orthogonal polynomials satisfying fourth order differential equations, Proc. Royal Soc. Edinburgh Sect. A 87 (1981), 271288. 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), 77102. MR 932778 (89e:34017)
 [10]
 H. L. Krall, Certain differential equations for Tchebycheff polynomials, Duke Math. J. 4 (1938), 705718. 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), 255265. MR 690030 (84c:42036)
 [13]
 , The Krall polynomials as solutions to a second order differential equation, Canad. Math. Bull. 26 (1983), 410417. MR 716580 (85c:33007)
 [14]
 , On the classification of differential equations having orthogonal polynomial solutions, Ann. Mat. Pura Appl. (4) 93 (1984), 3553. 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), 291295. 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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199412602023
PII:
S 00029947(1994)12602023
Article copyright:
© Copyright 1994
American Mathematical Society
