Infinite series involving harmonic numbers of convergence rate 1/2

2021 ◽  
pp. 1-21
Author(s):  
Wenchang Chu
Keyword(s):  
Convergence Rate ◽  
Infinite Series ◽  
Harmonic Numbers

Some evaluations of infinite series involving parametric harmonic numbers

2019 ◽  
Vol 15 (07) ◽  
pp. 1531-1546
Author(s):  
Ce Xu
Keyword(s):  
Infinite Series ◽  
Double Series ◽  
Asymptotic Formulas ◽  
Residue Theorem ◽  
Harmonic Numbers ◽  
Digamma Function ◽  
Special Cases

In this paper, we deduce the asymptotic formulas of parametric digamma function [Formula: see text] at the integers and poles. Then using these identities and residue theorem, we establish a large number of formulas of double series involving parametric harmonic numbers. Some illustrative special cases as well as immediate consequences of the main results are also considered.

Binomial sums involving harmonic numbers

2011 ◽  
Vol 61 (2) ◽  
Author(s):  
Marian Genčev
Keyword(s):  
Infinite Series ◽  
Rational Functions ◽  
Harmonic Numbers ◽  
Binomial Type ◽  
Hypergeometric Type ◽  
Number Π ◽  
Binomial Sums ◽  
Generalized Harmonic Numbers

AbstractThis paper develops the approach to the evaluation of a class of infinite series that involve special products of binomial type, generalized harmonic numbers of order 1 and rational functions. We give new summation results for certain infinite series of non-hypergeometric type. New formulas for the number π are included.

scholarly journals Supercongruences related to 3F2(1) involving harmonic numbers

2018 ◽  
Vol 14 (04) ◽  
pp. 1093-1109 ◽  
Author(s):  
Roberto Tauraso
Keyword(s):  
Infinite Series ◽  
Binomial Coefficients ◽  
Harmonic Numbers

We provide various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

scholarly journals New Multiple Harmonic Sum Identities

10.37236/3946 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Roberto Tauraso ◽  
Helmut Prodinger
Keyword(s):  
Infinite Series ◽  
Special Class ◽  
Partial Fraction ◽  
Harmonic Numbers ◽  
Elementary Method ◽  
Partial Fraction Decomposition ◽  
Binomial Sums

We consider a special class of binomial sums involving harmonic numbers and we prove three identities by using the elementary method of the partial fraction decomposition. Some applications to infinite series  and congruences are given.

q-Analogues of Guillera’s two series for π±2 with convergence rate 27 64

2020 ◽  
pp. 1-20
Author(s):  
Xiaojing Chen ◽  
Wenchang Chu
Keyword(s):  
Convergence Rate ◽  
Infinite Series ◽  
Classical Formula ◽  
Partial Sum

Three transformation formulae are established for the partial sum of Bailey’s well-poised [Formula: see text]-series. Their particular cases provide [Formula: see text]-analogues of Guillera’s two series for [Formula: see text] with convergence rate [Formula: see text], and for other classical [Formula: see text]-related infinite series.

scholarly journals A note on harmonic number identities, Stirling series and multiple zeta values

2019 ◽  
Vol 15 (07) ◽  
pp. 1323-1348
Author(s):  
Markus Kuba ◽  
Alois Panholzer
Keyword(s):  
Infinite Series ◽  
General Type ◽  
Multiple Zeta Values ◽  
Harmonic Numbers ◽  
Special Values ◽  
Multiple Zeta ◽  
Special Cases ◽  
Zeta Values ◽  
Finite Sums ◽  
Binomial Sums

We study a general type of series and relate special cases of it to Stirling series, infinite series discussed by Choi and Hoffman, and also to special values of the Arakawa–Kaneko zeta function, studied before amongst others by Candelpergher and Coppo, and also by Young. We complement and generalize earlier results. Moreover, we survey properties of certain truncated multiple zeta and zeta star values, pointing out their relation to finite sums of harmonic numbers. We also discuss the duality result of Hoffman, relating binomial sums and truncated multiple zeta star values.

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