Andrews-Beck Type Congruences Related to the Crank of a Partition

2021/01/26 by Shreejit Bandyopadhyay, Bandyopadhyay, Shreejit
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.10910

openalex publication_date 2021/01/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a suitable generating function, while the conjectures related to cranks were studied by Shane Chern using weighted partition moments. We revisit the conjectures on the crank of a partition by decomposing the relevant generating function and further explore connections with Apple-Lerch series and tenth order mock theta functions.

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