MacMahon's sums-of-divisors and their connection to multiple Eisenstein series

2023/12/11 by Bachmann, Henrik · 9 citations
#11F11 #11M32 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2312.06132

Abstract

We give explicit expressions for MacMahon's generalized sums-of-divisors q-series A_r and C_r by relating them to (odd) multiple Eisenstein series. Recently, these sums-of-divisors have been studied in the context of quasimodular forms, vertex algebras, N=4 SU(N) Super-Yang-Mills theory, and the study of congruences of partitions. We relate them to a broader mathematical framework and give explicit expressions for both q-series in terms of Eisenstein series and their odd variants.

Cited by

• MacMahon-type q-series
• Asymptotics of partition parts in arithmetic progressions
• Schur Eisenstein series and Schur MacMahon series
• q-Binomial expansions of the truncated MacMahon's q-series
• Applications of Faà di Bruno's formula to partition traces
• An identity for generating series of deformations of multiple zeta values within an algebraic framework
• A refined view of a curious identity for partitions into odd parts with designated summands
• Arithmetic properties of MacMahon-type sums of divisors: the odd case
• Quasimodularity and Limiting Behavior for Variations of MacMahon Series

Related

• MacMahon's sums-of-divisors and allied q-series
• Further study on MacMahon-type sums of divisors
• Recursive Formulas for MacMahon and Ramanujan q-series
• Arithmetic properties of MacMahon-type sums of divisors: the odd case
• An identity for generating series of deformations of multiple zeta values within an algebraic framework