Combinatorial Telescoping for an Identity of Andrews on Parity in Partitions

TL;DR

This paper develops a combinatorial telescoping method to prove partition identities, including Andrews' identity on q-little Jacobi polynomials, by classifying objects and establishing bijections.

Contribution

It introduces a novel combinatorial telescoping approach for partition identities and applies it to prove Andrews' identity using classification and involution techniques.

Findings

Established a bijective proof for MacMahon's classical identity.

Provided a combinatorial classification for triples of partitions related to Andrews' identity.

Demonstrated the effectiveness of telescoping methods in partition identity proofs.

Abstract

Following the method of combinatorial telescoping for alternating sums given by Chen, Hou and Mu, we present a combinatorial telescoping approach to partition identities on sums of positive terms. By giving a classification of the combinatorial objects corresponding to a sum of positive terms, we establish bijections that lead a telescoping relation. We illustrate this idea by giving a combinatorial telescoping relation for a classical identity of MacMahon. Recently, Andrews posed a problem of finding a combinatorial proof of an identity on the q-little Jacobi polynomials which was derived based on a recurrence relation. We find a combinatorial classification of certain triples of partitions and a sequence of bijections. By the method of cancelation, we see that there exists an involution for a recurrence relation that implies the identity of Andrews.