Rational extension of Newton diagram for the positivity of ${}_1{F}_2$ hypergeometric functions and Askey-Szegö problem

Abstract

We present a rational extension of Newton diagram for the positivity of ${}_1{F}_2$ generalized hypergeometric functions. As an application, we give upper and lower bounds for the transcendental roots $\beta (\alpha )$ of \begin{align*} \int_0^{j_{α, 2}} t^{-β} J_α(t) dt = 0\qquad(-1<α\le 1/2), \end{align*} where $j_{\alpha ,2}$ denotes the second positive zero of Bessel function $J_{\alpha }$.

23 pages, 6 figures