On identities of Watson type - Ballantine, Cristina (Ars mathematica contemporanea, 2019)
We prove several identities of the type ?$α (n) = Σ_{k=0}^∞ β (\frac{(n - k(k + 1)/2)} {2})$?. Here, the functions ?$α (n)$? and ?$β (n)$? count partitions with certain restrictions or the number of parts in certain partitions. Since G. N. Watson Proc. Lond. Math. Soc. (2) 42, 550-556 (1937) 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 M. Merca ''On the number of even parts in all partitions of $n$ into distinct parts'', Ann. Comb. (to appear) 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: bijective combinatorics; bijektivna kombinatorika; combinatorial identities; kombinatorične identitete; partitions; razčlenitve