Positive Integrals of Bessel Functions

Abstract

It is shown that some new and some already known positivity results for integrals of Bessel functions and for generalized hypergeometric functions can be easily obtained by writing the integrals and functions either as a sum of squares of Bessel functions with positive coefficients or as a fractional integral of such a sum. In particular, this method is used to prove that $\int_0^x {(x - t)^\lambda t^{\lambda + {1 / 2}} J_\alpha (t)dt > 0,\quad 0 \leqq \lambda \leqq \alpha - {1 / 2},\quad \alpha > 1/2, \quad x > 0,} $ and to give a simple proof of Steinig’s recent result that the Lommel function $S_{\mu ,\nu } (x) > 0$ for $x > 0$ if, $\mu = {1 / 2}$ and $ - {1 / 2} < \nu < {1 / 2}$, or if $\mu > {1 / 2}$ and $ - \mu \leqq \nu \leqq \mu $.