A GENERALIZATION OF BESSEL’S INTEGRAL FOR THE BESSEL COEFFICIENTS

Abstract

We derive an integral over the $m$-dimensional unit hypercube that generalizes Bessel’s integral for $J_n(x)$. The integrand is $G(x\psi(t)) \exp(−2\pi i n \cdot t)$, where $G$ is analytic, and $\psi(t) = e^{2\pi it_1} + \ldots + e^{2\pi it_m} + e^{−2\pi i(t_1+...+t_m)}$, while n is a set of non-negative integers. In particular, we consider the case when $G$ is a hypergeometric function $_{p}F_q$.