We give several different q-analogues of the following two congruences of Z.-W. Sun: $$\sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{8^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{2}{{{p^r}}}} \right)(\bmod {p^2})\;\text{and}\;} \sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{{16}^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{3}{{{p^r}}}} \right)(\bmod {p^2})} $$ is the Jacobi symbol. The proofs of them require the use of some curious q-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

中文翻译：

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$q$-Z.-W 的两个超同余的类似物。太阳

我们给出了 Z.-W 的以下两个同余的几个不同的 q 类比。太阳：$$\sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{8^k}}}\left( {\begin{array }{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{2}{{{p^r}}}} \right)( \bmod {p^2})\;\text{and}\;} \sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{ {16}^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{ 3}{{{p^r}}}} \right)(\bmod {p^2})} $$ 是雅可比符号。它们的证明需要使用一些奇怪的 q 系列恒等式，其中两个与富兰克林关于将分区划分为不同部分的对合有关。我们也在 2012 年证实了后一作者和曾某的猜想。