Summation formulae for binomial moments

Marta Na Chen; Wenchang Chu; Marta Na Chen; Wenchang Chu

doi:10.3934/era.2026286

Electronic Research Archive

2026, Volume 34, Issue 9: 6542-6557. doi: 10.3934/era.2026286

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Summation formulae for binomial moments

Marta Na Chen ^{1
,
,},
Wenchang Chu ^{2
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1.
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China
2.
Independent Researcher, Via Dalmazio Birago 9/E, Lecce 73100, Italy

Received: 15 April 2026 Revised: 17 July 2026 Accepted: 27 July 2026 Published: 31 July 2026

By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined:
$ \sum\limits_{k = 1}^n(\pm1)^{k-1} \binom{2n}{n-k}k^m \quad\text{and}\quad \sum\limits_{k = 1}^n(\pm1)^{k-1}\left[ \begin{array}{l}{{2n}}\ {{n-k}}\end{array} \right]k^m. $
Several explicit summation formulae are established with most of them having not appeared previously in the literature except for the first class corresponding to the "$ + $" sign.
- binomial moment,
- central binomial coefficient,
- telescoping method
Citation: Marta Na Chen, Wenchang Chu. Summation formulae for binomial moments[J]. Electronic Research Archive, 2026, 34(9): 6542-6557. doi: 10.3934/era.2026286

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Abstract

By combining the telescoping method with an algebraic relation, four classes of binomial moments are examined:

$ \sum\limits_{k = 1}^n(\pm1)^{k-1} \binom{2n}{n-k}k^m \quad\text{and}\quad \sum\limits_{k = 1}^n(\pm1)^{k-1}\left[ \begin{array}{l}{{2n}}\ {{n-k}}\end{array} \right]k^m. $

Several explicit summation formulae are established with most of them having not appeared previously in the literature except for the first class corresponding to the "$ + $" sign.

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