A generalization of multiple zeta values. Part 2: Multiple 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 200—233
DOI: 10.7546/nntdm.2022.28.2.200-233
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Authors and affiliations

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


Multiple zeta values have become of great interest due to their numerous applications in mathematics and physics. In this article, we present a generalization, which we will refer to as multiple sums, where the reciprocals are replaced with arbitrary sequences. We develop formulae to help with manipulating such sums. We develop variation formulae that express the variation of multiple sums in terms of lower order multiple sums. Additionally, we derive a set of partition identities that we use to prove a reduction theorem that expresses multiple sums as a combination of simple sums. We present a variety of applications including applications concerning polynomials and MZVs such as generating functions and expressions for \zeta(\{2p\}_m) and \zeta^\star(\{2p\}_m). Finally, we establish the connection between multiple sums and a type of sums called recurrent sums. By exploiting this connection, we provide additional partition identities for odd and even partitions.


  • Multiple sums
  • Viète’s formula
  • Polynomials
  • Generating function
  • Multiple zeta values
  • Riemann zeta function
  • Partitions
  • Stirling numbers
  • Bernoulli numbers
  • Faulhaber formula

2020 Mathematics Subject Classification

  • 11P84
  • 11B73
  • 11M32
  • 11C08


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Manuscript history

  • Received: 4 February 2021
  • Revised: 22 February 2022
  • Accepted: 12 April 2022
  • Online First: 18 April 2022

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Cite this paper

El Haddad, R. (2022). A generalization of multiple zeta values. Part 2: Multiple sums. Notes on Number Theory and Discrete Mathematics, 28(2), 200-233, DOI: 10.7546/nntdm.2022.28.2.200-233.

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