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Some series of squares of 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

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 1 , January 1930 , pp. 82 - 87
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* See G. N. Watson, Theory of Bessel Functions, § 16·14.

† Watson, loc. cit., § 5·41.

* See Whipple, F. J. W., “On well-poised series, etc,” Proc. London Math. Soc. (2), 24 (1925), 247–263, (5·2).Google Scholar Also Dougall, J., “On Vandermonde's theorem, and some more general expansions,” Proc. Edinburgh Math. Soc., 25 (1907), 114–132, (9).CrossRefGoogle Scholar

† Watson, loc. cit., §5·5.

‡ The justification is simpler if in (2·2) we omit the factors involving a 3, insert the factor (−1)r in the general term, and sum the resulting 4F 3(− 1).

§ Watson, loc. cit., § 11·41, (15), with φ = π.

* Watson, loc. cit., § 11·41, (15), with φ = ½ π.