Rational extension of Newton diagram for the positivity of ${}_1F_2$ hypergeometric functions and Askey-Szeg\"o problem

TL;DR

This paper extends the Newton diagram method rationally to analyze the positivity of ${}_1F_2$ hypergeometric functions and applies it to bound the roots of a Bessel function integral equation.

Contribution

It introduces a rational extension of the Newton diagram for ${}_1F_2$ functions and derives bounds for the roots of a Bessel integral equation.

Findings

Extended Newton diagram for ${}_1F_2$ functions.

Derived bounds for roots of Bessel integral equations.

Provided new insights into the positivity and root distribution of special functions.

Abstract

We present a rational extension of Newton diagram for the positivity of ${}_1F_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $\beta(\alpha)$ of \begin{align*} \int_0^{j_{\alpha, 2}} t^{-\beta} J_\alpha(t) dt = 0\qquad(-1<\alpha\le 1/2), \end{align*} where $j_{\alpha, 2}$ denotes the second positive zero of Bessel function $J_\alpha$.