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

In this paper, we establish a new asymptotic expansion of a ratio of two gamma functions, that is, as x → ∞, [Γ(x + u)/Γ(x + v)]1/(u − v) ~ (x + σ)exp [∑mk−1 B2m+1σ/wn(2n + 1) (x + σ)−2k + Rm (x; u, v)], where u,v ∈ with w = u−v ≠ 0 and ρ = (1−w) /2, σ = (u + v − 1) /2, B2n+1 (ρ) are the Bernoulli polynomials. We also prove that the function x → (−1)mRm (x;u,v) for m ∈ N is completely monotonic on \left (−\σ,∞) if |u−v| <1, which yields an explicit bound for |Rm (x;u,v)| and some new inequalities. [ABSTRACT FROM AUTHOR]