On identities of Watson type

Cristina Ballantine, Mircea Merca

Abstract


We prove several identities of the type α(n) = Σk=0nβ((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.


Keywords


Partitions, combinatorial identities, bijective combinatorics

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DOI: https://doi.org/10.26493/1855-3974.1782.127

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications