On Partial Sums of Mock Theta Functions of Order Five and Seven

Abstract

In a recent paper [3, 3.10-3.23, 6§] we have defined partial mock theta functions of order three and have shown how they are interrelated to each other.
The object of this paper is to define partial mock theta functions of orders 5 and 7. Unlike mock theta functions of order three, the mock theta functions of orders 5 and 7 have no direct basic hypergeometric definitions, although they have been shown to be limiting cases of certain ₃Φ₂ and ₄Φ₃ series, respectively, (see Anju Gupta [2, 152-161) and R.P. Agarwal [1, 99-101]). To avoid the limiting process we have defined the partial mock theta functions of orders five and seven as the partial series of the corresponding infinite series definitions, as given by Ramanujan.