The Newton Polyhedron and positivity of ${}_2F_3$ hypergeometric functions

TL;DR

This paper establishes conditions for the positivity of a specific hypergeometric function using Newton polyhedra and applies these results to determine regions where a related integral involving Bessel functions is non-negative.

Contribution

It introduces a novel approach using Newton polyhedra to determine positivity conditions for ${}_2F_3$ hypergeometric functions with positive parameters.

Findings

Derived sufficient conditions for positivity based on Newton polyhedra.

Identified extensive parameter regions where a Bessel-related integral is non-negative.

Abstract

As for the ${}_2F_3$ hypergeometric function of the form \begin{equation*} {}_2F_3\left[\begin{array}{c} a_1, a_2\\ b_1, b_2, b_3\end{array}\biggr| -x^2\right]\qquad(x>0), \end{equation*} where all of parameters are assumed to be positive, we give sufficient conditions on $(b_1, b_2, b_3)$ for its positivity in terms of Newton polyhedra with vertices consisting of permutations of $\,(a_2, a_1+1/2, 2a_1)\,$ or $\,(a_1, a_2+1/2, 2a_2).$ As an application, we obtain an extensive validity region of $(\alpha, \lambda, \mu)$ for the inequality \begin{equation*} \int_0^x (x-t)^{\lambda}\, t^{\mu} J_\alpha(t)\, dt \ge 0\qquad(x>0). \end{equation*}