On the Positivity of some ${}_1 F_2 $’s

Newton diagram of positivity for 1F2 generalized hypergeometric functions

“…For its positivity, Cooke [4] proved the case β = 0 , which is equivalent to (a, a + 1, 2a) ∈ P 1,2 (a > 0), (1.4) and Makai [10] proved the case β = −1/2, α > 1/2, that is, a, a + 1, 2a − 1 2 ∈ P 1,2 (a > 1). (1.5) (ii) In connection with completely monotone functions of certain type and the positivity of Cesáro means of Jacobi series, Askey and Pollard [2] and Fields and Ismail [6] proved separately a, 2a, 2a + 1 2 ∈ P 1,2 (a > 0), ( which includes (1.4), (1.6) as a special case δ = 1/2, a, respectively. In addition, as it will be explained below, Gasper invented a series expansion method for investigating positivity and obtained a number of positivity results for the Bessel integrals of certain type.…”

On Positive Definiteness of Some Functions

“…Proof. The sufficiency is shown by Fields, Ismail [10]. The necessity can be proved as follows: if x &$ (x 2 +1) &c # M (0, + ) , then it follows from (25) …”

“…In this case, the integral in the left-hand side exists because f b p is bounded and measurable. The Riemann Stieltjes integral in (10) does also exist since f # C [0, r] and p is measurable. Therefore, G is correctly defined and is an increasing function on [0, r].…”

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

“…which represents two symmetric infinite strips bounded by b + c = 3a + 1/2 and four half-lines parallel to the coordinate axes. By combining the methods of Fields and Ismail [11], Gasper [12] and fractional integrals with the squares of Bessel functions as kernels, two of the present authors established the following criterion. Owing to the relation of (1.6), it is easy to see that J α shares positive zeros in common with Bessel function J α and its square takes the form…”