Expansion of hypergeometric functions in series of other hypergeometric functions

Authors:
Yudell L. Luke and Richard L. Coleman

Journal:
Math. Comp. **15** (1961), 233-237

MSC:
Primary 33.20

DOI:
https://doi.org/10.1090/S0025-5718-1961-0123745-3

MathSciNet review:
0123745

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Abstract: In a previous paper [1] one of us developed an expansion for the confluent hypergeometric function in series of Bessel functions. A different expansion of the same kind given by Buchholz [2] was also studied. Since publication of [1], it was found that Rice [3] has also developed an expansion of this type, and yet a fourth expansion of this kind can be deduced from some recent work by Alavi and Wells [4]. In this note, we first deduce a multiplication formula for the Gaussian hypergeometric function which generalizes a statement of Chaundy, see (11), page 187 of [5], and includes a multiplication theorem for the confluent hypergeometric functions due to Erdélyi, see (7), page 283 of [5]. Our principal result is specialized to give an expansion of the confluent hypergeometric function in series of Bessel functions which includes the four above as special cases. With the aid of the Laplace transform, the latter result is used to derive an expansion of the Gaussian hypergeometric function in series of functions of the same kind with changed argument. This is advantageous since, throughout most of the unit disc, the change in argument leads to more rapidly converging series. For special values of the parameters, the expansion degenerates into known quadratic transformations.

**[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**, https://doi.org/10.1090/S0025-5718-1959-0107027-2**[2]**Herbert Buchholz,*Die konfluente hypergeometrische Funktion mit besonderer Berücksichtigung ihrer Anwendungen*, Ergebnisse der angewandten Mathematik. Bd. 2, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). MR**0054783****[3]**S. O. Rice,*Some properties of ₃𝐹₂(-𝑛,𝑛+1,𝜁;1,𝑝;𝑣)*, Duke Math. J.**6**(1940), 108–119. MR**0001412****[4]**Y. Alavi and C. P. Wells,*Expansions of parabolic wave functions*, Proc. Amer. Math. Soc.**10**(1959), 876–880. MR**0109232**, https://doi.org/10.1090/S0002-9939-1959-0109232-X**[5]**A. Erdélyi, et al.,*Higher Transcendental Functions*, v. 1, McGraw-Hill Book Company, Inc., 1953.**[6]**C. S. Meijer, ``Expansion theorems for the*G*-function,''*Indag. Math.*, (1952-1956), p. 14-18.**[7]**L. J. Slater,*Expansions of generalized Whittaker functions*, Proc. Cambridge Philos. Soc.**50**(1954), 628–631. MR**0064210****[8]**W. N. Bailey,*Generalized Hypergeometric Series*, Cambridge University Press, 1935.**[9]**A. Erdélyi, et al.,*Tables of Integral Transforms*, v. 1, McGraw-Hill Book Company, Inc., 1954, p. 196 and p. 215.**[10]**Yudell L. Luke,*On the computation of log𝑍 and 𝑎𝑟𝑐tan𝑍*, Math. Tables Aids Comput.**11**(1957), 16–18. MR**0084855**, https://doi.org/10.1090/S0025-5718-1957-0084855-1

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DOI:
https://doi.org/10.1090/S0025-5718-1961-0123745-3

Article copyright:
© Copyright 1961
American Mathematical Society