On Combinatorics and Congruences for q−Series

Abstract

This thesis is devoted towards the study of $q-$series, with specific emphasis on mock theta function treating it as $q-$series. The subject matter covers both combinatorial and related arithmetic concepts. The combinatorial techniques utilized in this context include Ferrers diagram, $(n+t)-$color partitions, bipartitions, and so forth. The Ferrers diagram is a well-established and efficacious combinatorial tool utilized for the graphical analysis of partitions. The renowned result related to partitions: `The number of partitions of $z$ into at most $n$ parts is equal to the number of partitions of $z$ into parts whose largest part is at most $n$' was proved with the help of Ferrers diagram. Here, we make use of Ferrers diagram by incorporating certain modifications. Next the $(n+t)-$color partitions (A. K. Agarwal and G. E. Andrews. Rogers-Ramanujan identities for partitions with ``N copies of N''. {\em Journal of Combinatorial Theory, Series A}, 45:40--49, 1987), this tool has led to many new insights in the field of combinatorics, by introducing alternative solution strategies. Bipartitions in which the set of partitions categorized in two classes of partitions, which is a novel approach to comprehensively study partitions in an elucidate manner. We provide combinatorial interpretations of some mock theta functions given by Ramanujan, Watson, Gordon and McIntosh, Gu and Hao. Further, we present the combinatorial interpretations for their generalized versions as well as for some of the $q-$series.

Secondly, we explore the arithmetic properties of some mock theta functions. The study encompasses congruences, parity results, and recurrence relations that establish connections between mock theta functions and certain restricted partition functions. We establish an infinite family of congruences. In addition, we investigate the matching coefficients pertaining to certain $q-$series. Our emphasis throughout this thesis is to demonstrate the application of Ramanujan's theta functions for the purpose of deriving identities, specifically Euler's pentagonal number theorem and the Jacobi triple product identity.

Lastly, we study the combinatorial aspects of congruence related to the partition function $a(z)$, which counts the number of partitions of $z$, where even parts can come with 2$-$colors and odd parts with 1$-$color. And we define an involution to give the combinatorial proof for the congruences of partition function $\mathsf{p}'_{\omega}(z)$ and $\mathsf{p}'_{\nu}(z)$ related to the third order mock theta functions $\omega(q)$ and $\nu(q)$, respectively.