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- Title
COMPLETE MONOTONICITY OF THE REMAINDER OF AN ASYMPTOTIC EXPANSION OF THE GENERALIZED GURLAND’S RATIO.
- Authors
ZHEN-HANG YANG; JING-FENG TIAN
- Abstract
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 B2k (θ1) - B2k (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 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results.
- Subjects
ASYMPTOTIC expansions; HYPERGEOMETRIC series; GAMMA functions; MONOTONIC functions
- Publication
Mathematical Inequalities & Applications, 2024, Vol 27, Issue 1, p231
- ISSN
1331-4343
- Publication type
Article
- DOI
10.7153/mia-2024-27-18