Rational Extension of the Newton Diagram for the Positivity of $${}_1F_2$$ 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 β(α) ofwhere j α,2 denotes the second positive zero of Bessel function J α .

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

“…A proof for the nonnegativity of Θ n under Saalschützian condition (4.1) and (A2) is given in our previous work [8,Lemma 3.1]. By tracking down the proof therein, it is not hard to find that the underlined condition of (A2) can be replaced by (4.2) due to the same reason as above.…”

“…Appendix. The following is a summary of our work [8], [9] concerning the positivity of 1 F 2 hypergeometric functions of similar type (see also [6], [7] for relevant applications and [15] for a probabilistic approach).…”

“…In our previous studies [8], [9], we considered the problem of positivity for 1 F 2 hypergeometric functions of similar type and described regions of positivity in the plane of denominator-parameter pairs by means of Newton polyhedra and their hyperbolic extensions (see Theorem 9.2 for a summary). In a consistent manner, we aim at providing sufficient conditions for the positivity of 2 F 3 hypergeometric functions of type (1.1) in terms of Newton polyhedra in the (b 1 , b 2 , b 3 )-space for each fixed (a 1 , a 2 ).…”

“…In our previous work [8], we exploited one of Whipple's transformation formulas and an induction argument to set up a criterion of positivity for the terminating 4 F 3 hypergeometric series of similar type. Concerning (1.5), however, as an appropriate version of Whipple's transformation formula is no longer available, we shall make use of a modification of more general transformation formulas developed by Fields and Wimp [12] to decompose it into a finite sum involving products of terminating 4 F 3 hypergeometric series, which will enable us to obtain a set of positivity criteria.…”

The zeros of certain Fourier transforms:Improvements of Pólya's results

“…It should be emphasized that the boundary limit condition f (1−) = 0 is not necessary for the positivity of U (x). For example, as a special case of Askey-Szegö problem, it is shown in [3,Theorem 4.5] that…”