Complete monotonicity of the remainder of an asymptotic expansion of the generalized Gurland's ratio (Q6619462)

The authors consider the generalized Gurland's ratio \N\[ \NQ_{a,b;c,d}(x):=\frac{\Gamma(x+a)\Gamma(x+b)}{\Gamma(x+c)\Gamma(x+d)} \N\] \Nfor the particular case \(a+b=c+d\). By using an appropriate integral representation of the logarithm of the gamma function, the authors derive an integral representation of the logarithm of the Gurland's ratio in the form o a Laplace transform. Then, they use Watson's lemma to derive the following asymptotic expansion of the logarithm of the Gurland's ratio when \(x\to\infty\), \N\[ \N\log\frac{\Gamma(x+a)\Gamma(x+b)}{\Gamma(x+c)\Gamma(x+d)}\sim\sum_{k=1}^\infty \frac{B_{2k}(\theta_1)-B_{2k}(\theta_2)}{k(2k-1)(x+r)^{2k-1}}, \N\] \Nwith \(r:=(a+b-1)/2=(c+d-1)/2\), \(\theta_k:=(1-\delta_k)/2\), \(\delta_1:=\vert a-b\vert\), \(\delta_2:=\vert c-d\vert\), and \(B_k(z)\) are the Bernoulli polynomials. Moreover, the remainder term of the expansion, \N\[ \NR_{n}(a,b;c,d;x):=\log \frac{\Gamma(x+a)\Gamma(x+b)}{\Gamma(x+c)\Gamma(x+d)}-\sum_{k=1}^n \frac{B_{2k}(\theta_1)-B_{2k}(\theta_2)}{k(2k-1)(x+r)^{2k-1}}, \N\] \Nis also considered. By using Bernstein theorem and some properties of the Bernoulli polynomials, the authors show that, when \(0\le\theta_2<\theta_1\le 1\), the function \((-1)^nR_{n}(a,b;c,d;x)\), as a function of \(x\), is completely monotonic on \((-r,\infty)\), and then, for \(x>-r\), \N\[ \N\vert R_{n}(a,b;c,d;x)\vert\le \frac{\vert B_{2n+2}(\theta_1)-B_{2n+2}(\theta_2)\vert}{(n+1)(2n+1)(x+r)^{2n+1}}. \N\] \NSeveral similar results are derived as corollaries for several particular interesting cases of the logarithm of the Gurland's ratio.