Double asymptotic inequalities for the generalized Wallis ratio

Asymptotic estimates for the generalized Wallis ratio \ (W^* (x): =1 { (x+1{2) } (x+1) \) are presented for \ (x^+\) on the basis of Stirling's approximation formula for the \ (\) function. For example, for an integer \ (p2\) and a real \ (x>-12\) we have the following double asymptotic inequality (p, x) \, <\, W^* (x) \, <\, B (p, x), \ where align* A (p, x): =& Wₚ (x) (1-18 (x+p) +1128 (x+p) ²+1379 (x+p) ³), \\ B (p, x): = & Wₚ (x) (1-18 (x+p) +1128 (x+p) ²+1191 (x+p) ³), \\ Wₚ (x): =& 1\, (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\).