Remarkable series concerning $  \binom{3n}{n} $ and harmonic numbers in numerators

Chunli Li; Wenchang Chu; Chunli Li; Wenchang Chu

doi:10.3934/math.2024837

AIMS Mathematics

2024, Volume 9, Issue 7: 17234-17258. doi: 10.3934/math.2024837

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Remarkable series concerning $ \binom{3n}{n} $ and harmonic numbers in numerators

Chunli Li ¹,
Wenchang Chu ^{2
,
,}

1.
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou (Henan), China
2.
Via Dalmazio Birago 9/E, Lecce 73100, Italy

Received: 26 March 2024 Revised: 11 May 2024 Accepted: 15 May 2024 Published: 20 May 2024
MSC : Primary 11B65; Secondary 33C20, 65B10

Three classes of infinite series containing binomial coefficient $ \binom{3n}{n} $, harmonic-like numbers, and an independent variable "$ y $" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the $ _3{F_2} $-series through the "coefficient extraction" method.
- Lambert's series,
- harmonic number,
- binomial coefficient,
- dilogarithm function,
- Bailey's cubic transformation
Citation: Chunli Li, Wenchang Chu. Remarkable series concerning $ \binom{3n}{n} $ and harmonic numbers in numerators[J]. AIMS Mathematics, 2024, 9(7): 17234-17258. doi: 10.3934/math.2024837

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Abstract

Three classes of infinite series containing binomial coefficient $ \binom{3n}{n} $, harmonic-like numbers, and an independent variable "$ y $" are examined. Several algebraic formulae in closed form are established, including, as special cases, three conjectured values for numerical series by Z.-W. Sun. This is fulfilled by integrating Lambert's series and manipulating the cubic transformations for the $ _3{F_2} $-series through the "coefficient extraction" method.

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