Multiple sums for cyclically symmetric products of binomial coefficients

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

doi:10.3934/math.20251285

AIMS Mathematics

2025, Volume 10, Issue 12: 29236-29262. doi: 10.3934/math.20251285

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Multiple sums for cyclically symmetric products of binomial coefficients

Marta Na Chen ^{1
,},
Wenchang Chu ^{2
,
,}

1.
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou, China; chennamarta@zknu.edu.cn
2.
Via Dalmazio Birago 9/E, Lecce 73100, Italy

Received: 17 September 2025 Revised: 27 November 2025 Accepted: 08 December 2025 Published: 12 December 2025
MSC : Primary 11B39, Secondary 05A15, 11B65

Carlitz' multiple sums involving cyclically symmetric products of binomial coefficients are extended by introducing weight monomials of $ m $-variables. By making use of recursive reductions and the Lagrange expansion formula, we determine, in closed form, the related rational generating functions that significantly generalize the classical result discovered by Carlitz in 1965. Several novel applications are presented as consequences.
- circular sum,
- generating function,
- binomial coefficient,
- Lucas number,
- Fibonacci number,
- Lagrange expansion formula
Citation: Marta Na Chen, Wenchang Chu. Multiple sums for cyclically symmetric products of binomial coefficients[J]. AIMS Mathematics, 2025, 10(12): 29236-29262. doi: 10.3934/math.20251285

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Abstract

Carlitz' multiple sums involving cyclically symmetric products of binomial coefficients are extended by introducing weight monomials of $ m $-variables. By making use of recursive reductions and the Lagrange expansion formula, we determine, in closed form, the related rational generating functions that significantly generalize the classical result discovered by Carlitz in 1965. Several novel applications are presented as consequences.

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