Newton diagram of positivity for ${}_1F_2$ generalized hypergeometric functions

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 F Let P a bethe 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.…”