Beck-type companion identities for Franklin's identity via a modular refinement

Abstract

The original Beck conjecture, now a theorem due to Andrews, states that the difference in the number of parts in all partitions into odd parts and the number of parts in all strict partitions is equal to the number of partitions whose set of even parts has one element, and also to the number of partitions with exactly one part repeated. This is a companion identity to Euler's identity. The theorem has been generalized by Yang to a companion identity to Glaisher's identity. Franklin generalized Glaisher's identity, and in this article, we provide a Beck-type companion identity for Franklin's identity and prove it via a modular refinement. We provide both analytical and combinatorial proofs. Andrews' and Yang's respective theorems fit naturally into this very general frame. We also give a generalization to Franklin's identity of the second Beck-type companion identity proved by Andrews and Yang in their respective work.