A generalization of multiple zeta values. Part 1: Recurrent sums

Roudy El Haddad
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 28, 2022, Number 2, Pages 167–199
DOI: 10.7546/nntdm.2022.28.2.167-199
Full paper (PDF, 340 Kb)

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Authors and affiliations

Roudy El Haddad
Engineering Department, La Sagesse University
Beirut, Lebanon

Abstract

Multiple zeta star values have become a central concept in number theory with a wide variety of applications. In this article, we propose a generalization, which we will refer to as recurrent sums, where the reciprocals are replaced by arbitrary sequences. We introduce a toolbox of formulas for the manipulation of such sums. We begin by developing variation formulas that allow the variation of a recurrent sum of order m to be expressed in terms of lower order recurrent sums. We then proceed to derive theorems (which we will call inversion formulas) which show how to interchange the order of summation in a multitude of ways. Later, we introduce a set of new partition identities in order to then prove a reduction theorem which permits the expression of a recurrent sum in terms of a combination of non-recurrent sums. Finally, we use these theorems to derive new results for multiple zeta star values and recurrent sums of powers.

Keywords

• Recurrent sums
• Partitions
• Multiple zeta star values
• Riemann zeta function
• Bell polynomials
• Stirling numbers
• Bernoulli numbers
• Faulhaber formula

2020 Mathematics Subject Classification

• 11P84
• 11B73
• 11M32
• 05A18

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Manuscript history

• Received: 3 February 2021
• Revised: 25 January 2022
• Accepted: 12 April 2022
• Online First: 18 April 2022