Combinatorial Telescoping for an Identity of Andrews on Parity in Partitions

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.