A Study on Product Formulae for Mock Theta Functions of Different Order

Abstract

Early in 1920, three months before his death, Ramanujan wrote his last letter to Hardy. In the course of it he said: I
discovered very interesting functions in recent times which I call ‘Mock θ- functions’. Unlike the ‘False θ-functions’, they
enter into mathematics as wonderfully as the ordinary θ- functions. I am sending you with this letter some examples. The
letter was accompanied by five foolscap passes. In the first three pages Ramanujan explained what he meant by a ‘Mock
θ- function’. Hardy’s comment about Mock θ- function is; A Mock θ- function is a function defined by a q-series,
convergent when |q|<1. We can calculate asymptotic formulae for it, when q tends to a rational point e 2πir/s of the unit
circle of the same degree of precision these furnished for the ordinary θ- function by the theory of linear transformation.