Study of Mock Theta Functions

Abstract

In this thesis, we studied some mock theta functions combinatorially defined by S. Ramanujan, three months before his death in 1920. We studied par- titions with n copies of n defined by Agarwal and Andrews in 1987 ( A.K. Agarwal and G.E. Andrews, Rogers-Ramanujan identities for partitions with "N copies of N", J. Combin. Theory, Ser. A, 45(1) (1987), 40-49 ) and lat- tice paths defined by A.K. Agarwal and Bressoud in 1989 (A.K. Agarwal and D.M. Bressoud, Lattice paths and multiple basic hypergeometric series, Pacific J. Math., 136(2) (1989), 209-228). Using these combinatorial objects we discussed combinatorial interpretation of four mock theta functions out of 17 mock theta functions defined by S. Ramanujan. Ramanujan asserted that these functions share certain proper- ties with theta functions, which have been investigated at great length by Carl Gustav Jacob Jacobi (1804-1851). Theta functions are essentially ellip- tic analogs of exponential function. Jacobi proved a variety of identities and found expressions for these special functions in terms of infinite series and infinite products. Our main results are contained in Chapters 2-3.