On the Positivity of some ₁ F₂’s

By using asymptotic methods and fractional integration, it is shown that \[ {}_1 F_2 ( {{array}{*{20}c} {λ - a} \\ {ρ λ + b,ρ λ + c} \\ {array} | {{{ - μ ^2 }}{4}} } ) 0, μ { real},\]0 a λ, 0 b, 1 2c, and either 2ρ 3, λ 1 or ρ 2, λ 0. From this, it is deduced that for $x > 0$, x2λ - 2ρ λ - b (1 - x² )- λ is completely monotonic for b 0 and either 2ρ 3, λ 1, or ρ 2, λ 0. This extends the results of [3] and proves some conjectures of Askey [1].