On identities of Watson type

Abstract

We prove several identities of the type α(n) = Σk=0n β((n − k(k + 1)/2) / 2). Here, the functions α(n) and β(n) count partitions with certain restrictions or the number of parts in certain partitions. Since Watson proved the identity for α(n) = Q(n), the number of partitions of n into distinct parts, and β(n) = p(n), Euler’s partition function, we refer to these identities as Watson type identities. Our work is motivated by results of G. E. Andrews and the second author who recently discovered and proved new Euler type identities. We provide analytic proofs and explain how one could construct bijective proofs of our results.

Locations

• Ars Mathematica Contemporanea - View - PDF