An equivalence theorem on generating functions

Author:
H. M. Srivastava

Journal:
Proc. Amer. Math. Soc. **52** (1975), 159-165

MSC:
Primary 33A70

DOI:
https://doi.org/10.1090/S0002-9939-1975-0379931-2

MathSciNet review:
0379931

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Abstract | References | Similar Articles | Additional Information

Abstract: The present paper establishes an equivalence theorem on generating relations for a sequence of functions defined by the Rodrigues formula (1) below. It is also shown how this theorem may be applied to a fairly large variety of special functions including, for instance, the classical orthogonal polynomials.

**[1]**F. Brafman,*Generating functions and associated Legendre polynomials*, Quart. J. Math. Oxford Ser. (2)**10**(1959), 156-160. MR**21**#6450. MR**0107728 (21:6450)****[2]**A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,*Higher transcendental functions*. Vol. II, McGraw-Hill, New York, 1953. MR**15**, 419.**[3]**H. L. Krall and O. Frink,*A new class of orthogonal polynomials*:*The Bessel polynomials*, Trans. Amer. Math. Soc.**65**(1949), 100-115. MR**10**, 453. MR**0028473 (10:453a)****[4]**G. Szegö,*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939. MR**1**, 14.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1975-0379931-2

Keywords:
Equivalence theorem,
generating relations,
Rodrigues' formula,
special functions,
classical orthogonal polynomials,
holomorphic functions,
the Cauchy-Hadamard theorem,
d'Alembert's ratio test,
Cauchy's integral formula,
uniform convergence,
binomial theorem,
analytic continuation,
Hermite polynomials,
Laguerre polynomials,
Jacobi polynomials,
Bessel polynomials,
ultraspherical polynomials,
Legendre polynomials,
Tchebycheff polynomials,
Charlier polynomials,
generating-function equivalence,
Meixner polynomials

Article copyright:
© Copyright 1975
American Mathematical Society