Classical Partition Functions And Congruences

By means of some basic identities for theta function, and the method of multi-section series, the authors investigate the identities for classical partition functions and congruence relations. The authors define multiranks for multipartitions and give the combinatorial interpretations of partition congruences due to Toh. Utilizing the theory of modular equations, the authors find several identities for partitions with distinct colors. The content is summarized as follows:(1) There are many important identities of the classical theta function such as Jacobi’s triple product identity, quintuple product identity, Weierstrass’three-term relation. Four equa-tions satisfied by Ramanujan’s cubic continued fraction are reviewed by means of Weier-strass’ three-term relation. By means of the q-series expansion, the authors also give some proofs of theta function identities. New proof for Chan’s cubic analogue of Ramanujan’s "most beautiful identity" about the generating function for p(5n+4) is presented.(2) By means of the multi-section series method, six congruence relations and their corre-sponding generating functions are established for three functions of multipartitions.(3) The authors generalize Hammond-Lewis birank to multiranks for partitions into colors and give combinatorial interpretations for the multipartition functions b(n) modulo5,7and Q(p1;p2)(n) modulo3,5introduced respectively by Zhao and Zhong in2011and Toh in2012(4) The authors establish sixteen identities for partitions with distinct colors that arise from Ramanujan’s formulas for multipliers in the theory of modular equations. Three of them are the conjectures made by Sandon and Zanello.