COMPLETE MONOTONICITY OF THE REMAINDER OF AN ASYMPTOTIC EXPANSION OF THE GENERALIZED GURLAND’S RATIO.

Let a,b,c,d ∈ R with a+b = c+d = 2r +1. Then In Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) ~ ∞/Σ/k=1 B_2k (θ₁) - B_2k (θ₂)/k (2k-1) (x+r)^2k-1 as x→∞, where (δ1,δ2) = (|a−b|,|c−d|) = (1−2θ₁,1−2θ₂). When 0 ≤ δ2 < δ1 ≤ 1, the function x→ (-1)^m [1n Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) - m/Σ/k=1 B_2k (θ₁) - B_2k (θ₂)/k (2k-1) (x+r)^2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results. [ABSTRACT FROM AUTHOR]