Double asymptotic inequalities for the generalized Wallis ratio

Asymptotic estimates for the generalized Wallis ratio \(W^*(x):={1 {√π}·{Γ(x+1/2)}{Γ(x+1)}\) are presented for \(x^+\) on the basis of Stirling's approximation formula for the \(Γ\) function. For example, for an integer \(p≥2\) and a real \(x>-1/2\) we have the following double asymptotic inequality \[A(p,x)\,<\,W^*(x)\,<\,B(p,x),\] where {align*} A(p,x):=& W_p(x)(1-1/8(x+p)+1/128(x+p)^2+1/379(x+p)^3), \\ B(p,x):= & W_p(x)(1-1/8(x+p)+1/128(x+p)^2+1/191(x+p)^3),\\ W_p(x):=& {1}{√π\,(x+p)}·{(x+1)⁽ᵖ⁾}{(x+1/2)⁽ᵖ⁾}, {align*} with \(y⁽ᵖ⁾≡ y(y+1)⋯(y+p-1)\), the Pochhammer rising (upper) factorial of order \(p\).