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Generating Functions for Bessel Functions

Published online by Cambridge University Press:  20 November 2018

Louis Weisner
Louis Weisner*
Affiliation:
University of New Brunswick
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On replacing the parameter n in Bessel's differential equation

1.1

by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where

1.2

This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.

It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 148 - 155
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Bailey, W. N., Generalized hyper geometric series (Cambridge, 1935).Google Scholar
2. Chaundy, T. W., An extension of hyper geometric functions, Quart. J. Math., 14 (1943), 55–78.Google Scholar
3. Watson, G. N., A treatise on Bess el functions (2nd ed., Cambridge, 1944).Google Scholar
4. Weisner, L., Group-theoretic origin of certain generating functions, Pacific J. Math. 4, supp. 2 (1955), 1033-9.Google Scholar
5. Weisner, L., Generating functions for Hermite functions, Can. J. Math., 11 (1959), 141–147.Google Scholar