The Gauss and Pfaff-Saalschütz theorems on classical hypergeometric series are employed to extend the summation formulae on Catalan numbers due to Jonah, Koshy and Touchard. Their q-analogs are also investigated by means of basic hypergeometric series.
We generalize Hammond-Lewis birank to multiranks for partitions into colors and give combinatorial interpretations for multipartitions such as $b(n)$ defined by H. Zhao and Z. Zhong and $Q_{p_1;p_2}(n)$ defined by Toh congruences modulo 3, 5, 7.
In a recent systematic study, C. Sandon and F. Zanello offered 30 conjectured identities for partitions. As a consequence of their study of partition identities arising from Ramanujan's formulas for multipliers in the theory of modular equations, the present authors in an earlier paper proved three of these conjectures. In this paper, we provide proofs for the remaining 27 conjectures of Sandon and Zanello. Most of our proofs depend upon known modular equations and formulas of Ramanujan for theta functions, while for the remainder of our proofs it was necessary to derive new modular equations and to employ the process of duplication to extend Ramanujan's catalogue of theta function formulas.
ABSTRACT. We establish several identities for partitions with distinct colors that arise from Ramanujan's formulas for multipliers in the theory of modular equations. In the course of our investigations, we prove three conjectures made by C. Sandon and F. Zanello. However, combinatorial proofs of these three conjectures and most of the theorems in this paper remain to be given.2010 Mathematics subject classifications: Primary 11P84; Secondary 05A15, 05A17.
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