Infinite series of convergence rate $ -1/4 $ about generalized harmonic numbers

Chunli Li; Wenchang Chu; Chunli Li; Wenchang Chu

doi:10.3934/era.2026262

Electronic Research Archive

2026, Volume 34, Issue 9: 5907-5940. doi: 10.3934/era.2026262

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Infinite series of convergence rate $ -1/4 $ about generalized harmonic numbers

Chunli Li ,
Wenchang Chu ^,

School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China

Received: 04 May 2026 Revised: 25 June 2026 Accepted: 08 July 2026 Published: 13 July 2026

Chu and Zhang (2014) found several remarkable transformations for well-poised hypergeometric series. One of them is employed in this paper to investigate infinite series of convergence rate "$ -1/4 $" involving generalized harmonic numbers. Numerous closed-form formulae are established that explicitly express a large class of series in terms of $ \pi $, $ \ln2 $, and the Hurwitz zeta function. Thirteen transformations are derived from harmonic series into nine seemingly simpler multifold $ k $-sums, whose exact values are still not determined.
- harmonic number,
- Gamma function,
- hypergeometric series,
- Hurwitz zeta function
Citation: Chunli Li, Wenchang Chu. Infinite series of convergence rate $ -1/4 $ about generalized harmonic numbers[J]. Electronic Research Archive, 2026, 34(9): 5907-5940. doi: 10.3934/era.2026262

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Abstract

Chu and Zhang (2014) found several remarkable transformations for well-poised hypergeometric series. One of them is employed in this paper to investigate infinite series of convergence rate "$ -1/4 $" involving generalized harmonic numbers. Numerous closed-form formulae are established that explicitly express a large class of series in terms of $ \pi $, $ \ln2 $, and the Hurwitz zeta function. Thirteen transformations are derived from harmonic series into nine seemingly simpler multifold $ k $-sums, whose exact values are still not determined.

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