Complete monotonicity of the remainder in an asymptotic series related to the psi function

Zhen-Hang Yang; Jing-Feng Tian

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 337-351
  • ISSN: 0011-4642

Abstract

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Let p , q ∈ ℝ with p - q ≥ 0 , σ = 1 2 ( p + q - 1 ) and s = 1 2 ( 1 - p + q ) , and let 𝒟 m ( x ; p , q ) = 𝒟 0 ( x ; p , q ) + ∑ k = 1 m B 2 k ( s ) 2 k ( x + σ ) 2 k , where 𝒟 0 ( x ; p , q ) = ψ ( x + p ) + ψ ( x + q ) 2 - ln ( x + σ ) . We establish the asymptotic expansion 𝒟 0 ( x ; p , q ) ∼ - ∑ n = 1 ∞ B 2 n ( s ) 2 n ( x + σ ) 2 n as x → ∞ , where B 2 n ( s ) stands for the Bernoulli polynomials. Further, we prove that the functions ( - 1 ) m 𝒟 m ( x ; p , q ) and ( - 1 ) m + 1 𝒟 m ( x ; p , q ) are completely monotonic in x on ( - σ , ∞ ) for every m ∈ ℕ 0 if and only if p - q ∈ [ 0 , 1 2 ] and p - q = 1 , respectively. This not only unifies the two known results but also yields some new results.

How to cite

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Yang, Zhen-Hang, and Tian, Jing-Feng. "Complete monotonicity of the remainder in an asymptotic series related to the psi function." Czechoslovak Mathematical Journal (2024): 337-351. <http://eudml.org/doc/299215>.

@article{Yang2024,
abstract = {Let $p,q\in \mathbb \{R\}$ with $p-q\ge 0$, $\sigma = \frac\{1\}\{2\} ( p+q-1)$ and $s=\frac\{1\}\{2\} ( 1-p+q)$, and let \[ \mathcal \{D\}\_\{m\} ( x;p,q ) =\mathcal \{D\}\_\{0\} ( x;p,q ) +\sum \_\{k=1\}^\{m\}\frac\{B\_\{2k\} ( s) \}\{2k ( x+\sigma ) ^\{2k\}\} , \] where \[ \mathcal \{D\}\_\{0\} ( x;p,q ) =\frac\{\psi ( x+p ) +\psi ( x+q ) \}\{2\}-\ln ( x+\sigma ) . \] We establish the asymptotic expansion \[ \mathcal \{D\}\_\{0\} ( x;p,q ) \sim -\sum \_\{n=1\}^\{\infty \} \frac\{B\_\{2n\} ( s ) \}\{2n ( x+\sigma ) ^\{2n\}\} \quad \text\{as\} \ x\rightarrow \infty , \] where $B_\{2n\} ( s ) $ stands for the Bernoulli polynomials. Further, we prove that the functions $( -1) ^\{m\}\mathcal \{D\}_\{m\} ( x;p,q )$ and $( -1) ^\{m+1\}\mathcal \{D\}_\{m\} ( x;p,q )$ are completely monotonic in $x$ on $( -\sigma ,\infty )$ for every $m\in \mathbb \{N\}_\{0\}$ if and only if $p-q\in [ 0, \tfrac\{1\}\{2\} ]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.},
author = {Yang, Zhen-Hang, Tian, Jing-Feng},
journal = {Czechoslovak Mathematical Journal},
keywords = {psi function; asymptotic expansion; complete monotonicity},
language = {eng},
number = {1},
pages = {337-351},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Complete monotonicity of the remainder in an asymptotic series related to the psi function},
url = {http://eudml.org/doc/299215},
year = {2024},
}

TY - JOUR
AU - Yang, Zhen-Hang
AU - Tian, Jing-Feng
TI - Complete monotonicity of the remainder in an asymptotic series related to the psi function
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 337
EP - 351
AB - Let $p,q\in \mathbb {R}$ with $p-q\ge 0$, $\sigma = \frac{1}{2} ( p+q-1)$ and $s=\frac{1}{2} ( 1-p+q)$, and let \[ \mathcal {D}_{m} ( x;p,q ) =\mathcal {D}_{0} ( x;p,q ) +\sum _{k=1}^{m}\frac{B_{2k} ( s) }{2k ( x+\sigma ) ^{2k}} , \] where \[ \mathcal {D}_{0} ( x;p,q ) =\frac{\psi ( x+p ) +\psi ( x+q ) }{2}-\ln ( x+\sigma ) . \] We establish the asymptotic expansion \[ \mathcal {D}_{0} ( x;p,q ) \sim -\sum _{n=1}^{\infty } \frac{B_{2n} ( s ) }{2n ( x+\sigma ) ^{2n}} \quad \text{as} \ x\rightarrow \infty , \] where $B_{2n} ( s ) $ stands for the Bernoulli polynomials. Further, we prove that the functions $( -1) ^{m}\mathcal {D}_{m} ( x;p,q )$ and $( -1) ^{m+1}\mathcal {D}_{m} ( x;p,q )$ are completely monotonic in $x$ on $( -\sigma ,\infty )$ for every $m\in \mathbb {N}_{0}$ if and only if $p-q\in [ 0, \tfrac{1}{2} ]$ and $p-q=1$, respectively. This not only unifies the two known results but also yields some new results.
LA - eng
KW - psi function; asymptotic expansion; complete monotonicity
UR - http://eudml.org/doc/299215
ER -

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