Four identities related to third-order mock theta functions

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. proved these identities by using q-series. In this paper, using some identities for the universal mock theta function g(x;q)=x-1-1+∑n=0∞qn2(x;q)n+1(qx-1;q)n,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^{2}}}{(x;q)_{n+1}(qx^{-1};q)_{n}}\right) , \end{aligned}$$\end{document}we provide different proofs of these four identities.