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

$$\begin{aligned} g(x;q)=x^{-1}\left( -1+\sum _{n=0}^{\infty }\frac{q^{n^{2}}}{(x;q)_{n+1}(qx^{-1};q)_{n}}\right) , \end{aligned}$$

we provide different proofs of these four identities.

中文翻译：

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与三阶模拟theta函数有关的四个恒等式

Ramanujan在他的《失落的笔记本》中为四阶模拟theta函数提供了四个标识。在2005年，借助复杂分析，Yesilyurt首次证明了这四个身份。最近，安德鲁斯等。通过使用q系列证明了这些身份。在本文中，对通用模拟theta函数使用一些标识

$$ \ begin {aligned} g（x; q）= x ^ {-1} \ left（-1+ \ sum _ {n = 0} ^ {\ infty} \ frac {q ^ {n ^ {2} }} {（x; q）_ {n + 1}（qx ^ {-1-1; q）_ {n}} \ right），\ end {aligned} $$

我们提供了这四个身份的不同证明。