Ramanujan’s mock theta functions

Abstract

In his last letter to G. H. Hardy, S. Ramanujan wrote, ” I discovered very interesting functions recently which I call mock theta functions”. He also gave a long list of ’third order’, ’fifth order’ and ’seventh order’ mock theta functions together with identities satisfied by them. All the results on third and fifth order mock theta functions were proved by G. N. Watson in his two celebrated papers [8,9]. Andrews rediscovered Ramanujan’s notebook while visiting Cambridge University and called it Ramanujan’s ’lost’ notebook. The ’lost’ notebook contains several further results on mock theta functions which were proved by Andrews [1,2], Andrews and Garvan [3], Andrews and Hickerson [4]. Lastly the ’lost’ notebook contains eight identities for the tenth order mock theta function. Choi [5] has proved the first two of Ramanujan’s tenth order mock theta function identities and said that further identities will be proved in subsequent papers. In this paper we have given relations and expansions of partial mock theta functions, mock theta functions of tenth, third, fifth and sixth order. We have also given two Continued Fractions for tenth order mock theta functions. In section 4, we give a proof of a simple identity. In section 5, using this identity we connect the tenth order mock theta functions, partial tenth order mock theta functions with mock theta functions, partial mock theta functions of third, fifth, and sixth order. In section 6, we give relations between fifth order mock theta functions and third order mock theta functions and their partial sums. In section 7, we have given expansions of a tenth order mock theta functions in terms of partial mock theta function of tenth order. In section 8, we have proved two lemmas and with the help of these lemmas expressed the tenth order mock theta functions as a Continued Fraction.