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On a second order mock theta functions *.(Report)

In this paper we consider a second order mock theta function, recently given by Hikami. We define a generalized function and show it is a q [F.sub.q]-function and give its integral representation and multibasic expansion. We also show that this mock theta function can be obtained by performing a left half-shift transformation on a certain theta series.

Keywords and Phrases: Mock theta functions, Basic hypergeometric series.

1. Introduction

Recently Hikami (3) introduced the q-series[D.sub.5][q],

[D.sub.5][q] = [[infinity].summation over.(n=0)]][(-q;q).sub.n][q.sup.n]/(q;[q.sup.2]).sub.n + 1]], where q = [e.sup.2[pi]i[tau]]

in connection with the quantum invariant of ...

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