The Newton Polyhedron and positivity of ${}_2{F}_3$ hypergeometric functions

Abstract

As for the ${}_2{F}_3$ 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)^λ\, t^μ J_α(t)\, dt \ge 0\qquad(x>0). \end{equation*}

The paper is accepted to <Constructive Approximation>