Inequalities, asymptotic expansions and complete monotonicity properties related to the gamma and digamma functions

The quotient of two gamma functions has the following known asymptotic expansion: \(\displaystyle \frac{\Gamma(x+a)}{\Gamma(x+b)}\sim x^{a-b}\sum_{k=0}^{\infty}\binom{a-b}{k}B_{k}^{(a-b+1)}(a)\,x^{-k}\quad(x\to\infty),\) where \(B_{k}^{(\nu)}(x)\) denotes the generalized Bernoulli polynomials. In this article, without using these generalized Bernoulli polynomials \(B_{k}^{(\nu)}(x)\), we present two asymptotic expansions and a recurrence relation for determining the coefficients of each asymptotic expansion related to the quotient \(\frac{\Gamma(x+a)}{\Gamma(x+b)}\) of two gamma functions. Let \(\displaystyle \Omega_{n}:=\frac{\pi^{n/2}}{\Gamma\left(\frac{n}{2}+1\right)},\quad n\in\mathbb{N}\) be the volume of the unit ball in \(\mathbb{R}^{n}\). We establish a sharp two-sided inequality for the quotient \(\frac{\Omega_{n-1}}{\Omega_{n}}\). We also present a complete monotonicity property for the Nielsen's \(\beta\)-function, which extends a known result due to Matejíčka.