Double asymptotic inequalities for the generalized Wallis ratio

Abstract

Asymptotic estimates for the generalized Wallis ratio \(W^*(x):=\frac{1 {\sqrt{\pi}}\cdot\frac{\Gamma(x+\frac{1}{2})}{\Gamma(x+1)}\) are presented for \(x\in\mathbb{R}^+\) on the basis of Stirling's approximation formula for the \(\Gamma\) function. For example, for an integer \(p\ge2\) and a real \(x>-\tfrac{1}{2}\) we have the following double asymptotic inequality \[A(p,x)\,<\,W^*(x)\,<\,B(p,x),\] where \begin{align*} A(p,x):=& W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{379(x+p)^3}\right), \\ B(p,x):= & W_p(x)\left(1-\tfrac{1}{8(x+p)}+\tfrac{1}{128(x+p)^2}+\tfrac{1}{191(x+p)^3}\right),\\ W_p(x):=& \frac{1}{\sqrt{\pi\,(x+p)}}\cdot\frac{(x+1)^{(p)}}{(x+\frac{1}{2})^{(p)}}, \end{align*} with \(y^{(p)}\equiv y(y+1)\cdots(y+p-1)\), the Pochhammer rising (upper) factorial of order \(p\).