Combinatorics of Mock Theta Functions and q-Series

Abstract

This thesis predominantly studies mock theta functions combinatorially. However, there are some chapters dwelling into the combinatorics of $q$-series also. The combinatorial tools employed in these studies are $(n+t)$--color partitions (A. K. Agarwal and G. E. Andrews. Rogers-Ramanujan identities for partitions with ``N copies of N'', Journal of Combinatorial Theory, Series A, 45:40--49, 1987), generalized Frobenius partitions (G. E. Andrews. Generalized Frobenius partitions, American Mathematical Society, 301, 1984) and lattice paths (A.K. Agarwal and D. M. Bressoud. Lattice paths and multiple basic hypergeometric series, Pacific Journal of Mathematics, 136:209--228, 1989). Here, we find interpretations of twenty five mock theta functions by employing above mentioned tools and a novel idea of attaching weights to the partitions generated by the unsigned version of mock theta functions. To obtain interpretations of some mock theta functions in terms of lattice paths, the terminology of paths given by Agarwal and Bressoud has been modified by introducing backward horizontal steps. With these modifications the formed lattice paths naturally correspond to the $n$-color compositions. In addition to above, we provide combinatorial interpretations of some generalized $q$-series. Firstly, combinatorial interpretations of seven generalized $q$-series are obtained which have been earlier interpreted in terms of split $(n+t)$--color partitions. Secondly, we explore the concept of hook differences (Andrews, G. E., Baxter, R., Bressoud, D. M., Burge, W. H., Forrester, P. and Viennot, G. Partitions with prescribed hook differences, European Journal of Combinatorics, 8(4):341 350, 1987) which led to the generalization of the successive rank theorem to an identity involving partitions with prescribed hook differences. This identity involves a complex product but in some particular cases, it reduces to a simple triple or quintuple product. Our investigation in this thesis shows that one of these reductions involving triple product provides $(n+t)$--color analogue of Gordon's theorem which is also a generalization of the result of Agarwal and Andrews (A. K. Agarwal and G. E. Andrews. Rogers-Ramanujan identities for partitions with ``N copies of N'', Journal of Combinatorial Theory, Series A, 45:40--49, 1987). A similar manipulation of another such reduction involving quintuple product provides $(n+t)$--color partitions for a quintuple product, that initially arose in an identity due to Sills.