Complete monotonicity of the remainder in an asymptotic series related to the psi function,Czechoslovak Mathematical Journal

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Complete monotonicity of the remainder in an asymptotic series related to the psi function
Czechoslovak Mathematical Journal ( IF 0.5 ) Pub Date : 2024-02-12 , DOI: 10.21136/cmj.2024.0354-23
Zhen-Hang Yang , Jing-Feng Tian

Let p, q ∈ ℝ with p − q ≽ 0, $\sigma = {1 \over 2}(p + q - 1)$ and $s = {1 \over 2}(1 - p + q)$, and let

$${{\cal D}_m}(x;p,q) = {{\cal D}_0}(x;p,q) + \sum\limits_{k = 1}^m {{{{B_{2k}}(s)} \over {2k{{(x + \sigma )}^{2k}}}},} $$

where

$${{\cal D}_0}(x;p,q) = {{\psi (x + p) + \psi (x + q)} \over 2} - \ln (x + \sigma ).$$

We establish the asymptotic expansion

$${{\cal D}_0}(x;p,q) \sim - \sum\limits_{n = 1}^\infty {{{{B_{2n}}(s)} \over {2n{{(x + \sigma )}^{2n}}}}\,\,\,\,\,{\rm{as}}\,\,x \to \infty ,} $$

where B_2n(s) stands for the Bernoulli polynomials. Further, we prove that the functions ${( - 1)^m}{{\cal D}_m}(x;p,q)$ and ${( - 1)^{m + 1}}{{\cal D}_m}(x;p,q)$ are completely monotonic in x on (−σ, ∞) for every m ∈ ℕ₀ if and only if $p - q \in [0,{1 \over 2}]$ and p − q = 1, respectively. This not only unifies the two known results but also yields some new results.

更新日期：2024-02-12

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