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Published online by Cambridge University Press: 24 October 2008
The object of this note is to prove the following results, all of which hold when |a| < 1.
(2) If r is any positive integer other than zero, the
* Equations (1) and (3) were suggested for positive values of a(0<a<l) by some theorems of mine in null series (Proc. London Math. Soc. (2), 26 (1927), p. 54, Theorems 2 and 3). These theorems can be applied to transform Kapteyn series into integrals, but the proofs are very elaborate. For the proof below I am very much indebted to the referee. Apart from its simplicity it is true for complex values of a in the more extensive region |a| < 1.Google Scholar
† Watson, , Theory of Bessel Functions, p. 147 (5).Google Scholar
* Bromwich, , Theory of infinite series, 2nd Ed., p. 500, Theorem B.Google Scholar