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Transformations of Basic Hypergeometric Functions of any Order

Transformations of Basic Hypergeometric Functions of any Order 1948] BASIC HYPERGEOMETBIC FUNCTIONS 181 TRANSFORMATIONS OF BASIC HYPERGEOMETRIC FUNCTIONS OF ANY ORDER By D. B. SEARS [Received 22 May 1948.—Read 17 June 1948] Introduction 1. In a recent paperf I have shown that the hypergeometric function F(a ...,a ; b...,b ) v M+1 v M may be expressed in terms of M similar functions, and tha t this relation is a particular case of a general theorem from which may be derived trans- formations of trigonometric series. I consider now the corresponding problem for basic functions, using precisely the same methods as before. The main differences in the results obtained are due to the fact that, whereas before it was necessary to con- sider the function F(a ,...,a ; b ...,b ; x) on the circle \x\ = 1, for 1 M+1 v M basic series O(a ...,a ; b ,...,b \ x) the results are valid in the annulus lJ 3f+1 x M \pa | < | re | < 1, where p is the base, and a = b...b/a...a . 1 M 1 M+1 Transformations for the basic function of any order have been obtained by Watson, J who used contour integrals of Barnes's type for http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Proceedings of the London Mathematical Society Wiley

Transformations of Basic Hypergeometric Functions of any Order

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6115
eISSN
1460-244X
DOI
10.1112/plms/s2-53.3.181
Publisher site
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Abstract

1948] BASIC HYPERGEOMETBIC FUNCTIONS 181 TRANSFORMATIONS OF BASIC HYPERGEOMETRIC FUNCTIONS OF ANY ORDER By D. B. SEARS [Received 22 May 1948.—Read 17 June 1948] Introduction 1. In a recent paperf I have shown that the hypergeometric function F(a ...,a ; b...,b ) v M+1 v M may be expressed in terms of M similar functions, and tha t this relation is a particular case of a general theorem from which may be derived trans- formations of trigonometric series. I consider now the corresponding problem for basic functions, using precisely the same methods as before. The main differences in the results obtained are due to the fact that, whereas before it was necessary to con- sider the function F(a ,...,a ; b ...,b ; x) on the circle \x\ = 1, for 1 M+1 v M basic series O(a ...,a ; b ,...,b \ x) the results are valid in the annulus lJ 3f+1 x M \pa | < | re | < 1, where p is the base, and a = b...b/a...a . 1 M 1 M+1 Transformations for the basic function of any order have been obtained by Watson, J who used contour integrals of Barnes's type for

Journal

Proceedings of the London Mathematical Society – Wiley

Published: Jan 1, 1951

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