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 \(p\).