Newton diagram of positivity for ₁F₂ generalized hypergeometric functions

Abstract

As for the positivity of 1 F 2 generalized hypergeometric functions, we present a list of necessary and sufficient conditions in terms of parameters and determine the region of positivity by certain Newton diagram.

An extension of positivity for integrals of Bessel functions and Buhmann’s radial basis functions

“…To investigate the sign of 1 F 2 generalized hypergeometric function Φ, we shall make use of the following general criterion recently established by the authors [5], which will be applied subsequently in other occasions as well.…”

Rational Extension of the Newton Diagram for the Positivity of $${}_1F_2$$ Hypergeometric Functions and Askey–Szegö Problem

“…In this section we aim to extend the aforementioned positivity criterion of [8] for the generalized hypergeometric functions of type (1.8).…”

“…with parameters a > 0, b > 0, c > 0. In the recent work [8], to be explained in detail, a positivity criterion for the functions of type (1.8) is established in terms of the Newton diagram associated to {(a + 1/2, 2a), (2a, a + 1/2)}. Due to certain region of parameters left undetermined, however, it turns out that an application of the criterion to (1.7) yields Theorem A immediately but does not cover the missing region T either.…”

“…(Cho and Yun,[8]) For a > 0, b > 0, c > 0, putΦ(x) = 1 FLet P a be the Newton diagram associated to Λ = a + 1 2 , 2a , 2a, a + 1 O a the set defined in (4.1),(4.2) and N a the complement of P a ∪ O a in R so that the decomposition R 2 + = P a ∪ O a ∪ N a holds. (i) If Φ ≥ 0, then necessarily b > a, c > a, b + c ≥ 3a If (b, c) ∈ P a , then Φ ≥ 0 and strict positivity holds unless (b, c) ∈ Λ.…”

An Extension of the Multiple Erdélyi-Kober Operator and Representations of the Generalized Hypergeometric Functions

“…The nonnegativity of the G function in the integrand combined with γ − 1 ≥ 0 completes the proof. In a nice recent paper [5] the authors found the exact range of positive parameters α, β 1 , β 2 that ensure the inequality 1 F 2 (α; β 1 , β 2 ; x) ≥ 0 for all real x. This range can be described as follows: for α > 0 let P α denote the convex hull of the points (α m , ∞), (α m , α M ), (α M , α m ), (∞, α m ) in the plane (β 1 , β 2 ), where α m = min(2α, α + 1/2), α M = max(2α, α + 1/2).…”