Expansions of hypergeometric functions in hypergeometric functions

Authors:
Jerry L. Fields and Jet Wimp

Journal:
Math. Comp. **15** (1961), 390-395

MSC:
Primary 33.20

MathSciNet review:
0125992

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Abstract: In [1] Luke gave an expansion of the confluent hypergeometric function in terms of the modified Bessel functions . The existence of other, similar expansions implied that more general expansions might exist. Such was the case. Here multiplication type expansions of low-order hypergeometric functions in terms of other hypergeometric functions are generalized by Laplace transform techniques.

**[1]**Yudell L. Luke,*Expansion of the confluent hypergeometric function in series of Bessel functions*, Math. Tables Aids Comput.**13**(1959), 261–271. MR**0107027**, 10.1090/S0025-5718-1959-0107027-2**[2]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Higher Transcendental Functions*, McGraw-Hill Book Company, Inc., 1953, Vol. 1. Hereafter in this list, we use the abbreviation*H.T.F.***[3]**Yudell L. Luke and Richard L. Coleman,*Expansion of hypergeometric functions in series of other hypergeometric functions*, Math. Comp.**15**(1961), 233–237. MR**0123745**, 10.1090/S0025-5718-1961-0123745-3**[4]**A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi,*Tables of Integral Transforms*, Vol. 1, McGraw-Hill Book Company, Inc., 1954.**[5]**E. D. Rainville,*Certain generating functions and associated polynomials*, Amer. Math. Monthly**52**(1945), 239–250. MR**0011751****[6]**T. W. Chaundy,*An extension of hypergeometric functions. I*, Quart. J. Math., Oxford Ser.**14**(1943), 55–78. MR**0010749****[7]**C. S. Meijer, ``Expansion theorems for the G-function,''*Indag. Math.*, v. 14-17, 1952-55.**[8]**H.T.F., Vol. 2.**[9]**J. L. Fields & Y. L. Luke, ``Asymptotic expansions of a class of hypergeometric polynomials with respect to the order,'' Midwest Research Institute Technical Report, July 1, 1959.**[10]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110**

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1961-0125992-3

Article copyright:
© Copyright 1961
American Mathematical Society