A new asymptotic expansion of a ratio of two gamma functions and complete monotonicity for its remainder

Abstract

In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as x → ∞ x\rightarrow \infty , [ Γ ( x + u ) Γ ( x + v ) ] 1 / ( u − v ) ∼ ( x + σ ) exp ⁡ [ ∑ k = 1 m B 2 n + 1 ( ρ ) w n ( 2 n + 1 ) ( x + σ ) − 2 k + R m ( x ; u , v ) ] , \begin{equation*} \left [ \frac {\Gamma \left ( x+u\right ) }{\Gamma \left ( x+v\right ) }\right ] ^{1/\left ( u-v\right ) }\thicksim \left ( x+\sigma \right ) \exp \left [ \sum _{k=1}^{m}\frac {B_{2n+1}\left ( \rho \right ) }{wn\left ( 2n+1\right ) }\left ( x\!+\!\sigma \right ) ^{-2k}\!+\!R_{m}\left ( x;u,v\right ) \right ] , \end{equation*} where u , v ∈ R u,v\in \mathbb {R} with w = u − v ≠ 0 w=u-v\neq 0 and ρ = ( 1 − w ) / 2 \rho =\left ( 1-w\right ) /2 , σ = ( u + v − 1 ) / 2 \sigma =\left ( u+v-1\right ) /2 , B 2 n + 1 ( ρ ) B_{2n+1}\left ( \rho \right ) are the Bernoulli polynomials. We also prove that the function x ↦ ( − 1 ) m R m ( x ; u , v ) x\mapsto \left ( -1\right ) ^{m}R_{m}\left ( x;u,v\right ) for m ∈ N m\in \mathbb {N} is completely monotonic on ( − σ , ∞ ) \left ( -\sigma ,\infty \right ) if | u − v | > 1 \left \vert u-v\right \vert >1 , which yields an explicit bound for | R m ( x ; u , v ) | \left \vert R_{m}\left ( x;u,v\right ) \right \vert and some new inequalities.