Four identities related to third order mock theta functions

Abstract

Ramanujan presented four identities for third order mock theta functions in his Lost Notebook. In 2005, with the aid of complex analysis, Yesilyurt first proved these four identities. Recently, Andrews et al. provided different proofs by using $q$-series. In this paper, in view of some identities of a universal mock theta function \begin{align*} g(x;q)=x^{-1}\left(-1+\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(x;q)_{n+1}(qx^{-1};q)_{n}}\right), \end{align*} we establish new proofs of these four identities. In particular, by means of an identity of $g(x;q)$ given by Ramanujan and some theta function identities due to Mortenson, we find a new simple proof of the fourth identity.