Ramanujan-Type Congruences For Several Partition Functions

Based on the previous work on the basic hypergeometric series of Ramanujan, Chu, Baruah, Ojah et al., this thesis studies the Ramanujan-type congruences systematically, which is one im-portant congruence relation of the basic hypergeometric series, by means of the Congruence lemma on partition function, Jacobi identities, theta function identities. Several partition func-tions are investigated and some interesting Ramanujan-type congruences are established during this thesis. The content can be summarized as follows:Chapter1, research advance on the the basic hypergeometric series and Ramanujan-type congruences are reviewed briefly, and the new research direction of this thesis is analyzed.Chapter2, the basic knowledge and properties of the basic hypergeometric series and par-tition functions are proposed. Then present the new notions and combinational explanations of the relative partition functions.Chapter3, recall one method of proving Ramanujan’s three congruences, which is given mainly based on the Congruence lemma on partition function, Euler’s pentagon number theo-rem, Jacobi triple product identity.Chapter4, inspired by the recent work of Baruah and Ojah, several partition functions are constructed. Using the method involved in the last chapter and theta function identities, the re-sponding analogues of Ramanujan identities and some interesting Ramanujan-type congruences are obtained.