Evaluations of some Euler-Apéry-type series
type: Item
instance of: scholarly article
title: Evaluations of some Euler-Apéry-type series
author: Yujie Wang
author: Ying Li
author: Ce Xu
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 2022-11-09T00:00:00Z
review text: The subject of this paper is the Euler-Apéry-type series whose general terms is a product of central binomial coefficients, generalized harmonic numbers and \((n 4^n)^{-1}\). In particular, the authors use the method of contour integration to prove the result that the Euler-Apéry-type series \[ \sum_{n=1}^\infty \frac{n}{(n-1/2)^q}\cdot \frac{\binom{2n}{n}}{4^n} \in \mathbb{Q}[\pi, \log 2, \zeta(3), \zeta(5), \zeta(7), \ldots]. \] Furthermore, they give an explicit evaluation for the EulerApéry-type series \[ \sum_{n=1}^\infty \binom{mn}{n}\frac{x^n}{n^p} \] for positive integers \(m\) and \(p\), by using the method of generating function involving Fuss-Catalan numbers. Finally, they establish a recurrence relation for general Euler-Apéry-type series involving multiple harmonic star sum.
reviewed by: Sami Omar
Mathematics Subject Classification ID: 11M32
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 11B65
zbMATH DE Number: 7613966
zbMATH Keywords: Euler-Apéry-type series
zbMATH Keywords: contour integration
zbMATH Keywords: Fuss-Catalan numbers
zbMATH Keywords: generating function
describes a project that uses: MultipleZetaValues
MaRDI profile type: Publication
full work available at URL: s13226-021-00191-9
OpenAlex ID: W3203624579
cites work: Q4184941
cites work: Multivariate Fuss-Catalan numbers
cites work: Q3465227
cites work: Algebras of Iterated Path Integrals and Fundamental Groups
cites work: Iterated path integrals
cites work: Generalized Arakawa-Kaneko zeta functions
cites work: Q4769056
cites work: Euler Sums and Contour Integral Representations
cites work: Q5717207
cites work: Catalan numbers, their generalization, and their uses
cites work: Interesting Series Involving the Central Binomial Coefficient
cites work: Multiple harmonic series
cites work: Alternating multiple zeta values, and explicit formulas of some Euler-Apéry-type series
cites work: Identities for the multiple zeta (star) values
cites work: Explicit relations between multiple zeta values and related variants
cites work: Q4324348
cites work: Analytic continuation of multiple polylogarithms
cites work: Multiple zeta functions, multiple polylogarithms and their special values
DOI: 10.1007/S13226-021-00191-9
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description: scientific article; zbMATH DE number 7613966