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A note on Bateman's expansion in Bessel functions

Published online by Cambridge University Press:  24 October 2008

W. N. Bailey
W. N. Bailey
Affiliation:
Trinity College
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Extract

1. The following proof of Bateman's expansion may be of interest since it avoids the use of Appell's hypergeometric functions of two variables, though it suffers from the defect that it presupposes a knowledge of the expansion.

Type
Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 25 , Issue 1 , January 1929 , pp. 48 - 49
Copyright
Copyright © Cambridge Philosophical Society 1929

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References

* Watson, G. N., Theory of Bessel Functions, §11·6.Google Scholar

† See Watson, , loc. cit., § 4·42. The use of the symbol is necessary, since one of the parameters in the denominator is a negative integer.Google Scholar

* There is no great difficulty in justifying the implied change in the order of summation. The argument is similar to that used by Watson, loc. cit., p. 400.

† Whipple, F. J. W., “On well-poised series, etc”, Proc. London Math. Soc. (2), 24 (1925), 247–263, formula (6·3).Google ScholarSee also Whipple, , “A fundamental relation between generalized hypergeometric series,” Journal London Math. Soc., 1 (1926), 138–145, §2.CrossRefGoogle Scholar

‡ See, e.g. Hardy, G. H., “A chapter from Ramanujan's note-book,” Proc. Camb. Phil. Soc., 21 (1923), 492–503, formula (5·2).Google Scholar