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

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

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]**Y. L. Luke, ``Expansion of the confluent hypergeometric function in series of Bessel functions,''*MTAC*, v. 13, 1959, p. 261-271. MR**0107027 (21:5754)****[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]**Y. L. Luke & Richard L. Coleman, ``Expansion of hypergeometric functions in series of other hypergeometric functions,''*Math. Comp.*, v. 15, 1961, p. 233. MR**0123745 (23:A1067)****[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 their associated polynomials,''*Amer. Math. Monthly*, v. 52, No. 5, May 1945, or H.T.F., Vol. 3, p. 239-250. MR**0011751 (6:211e)****[6]**T. W. Chaundy, ``An extension of hypergeometric functions,''*Quart. J. Math. Oxford Ser.*14, 1943, p. 55-78. MR**0010749 (6:64d)****[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 University Press, 1945. MR**1349110 (96i:33010)**

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

Article copyright:
© Copyright 1961
American Mathematical Society