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We give -analogues of the following two Ramanujan-type formulas for : \begin{align*} \sum_{k=0}^\infty (6k+1)\frac{(\frac{1}{2})k^3}{k!^3 4^k} =\frac{4}{\pi} \quad\quad \sum{k=0}^\infty (-1)^k(6k+1)\frac{(\frac{1}{2})_k^3}{k!^3 8^k } =\frac{2\sqrt{2}}{\pi}. \end{align*} Our proof is based on two -WZ pairs found by the first author in his earlier work.

生成讲解博客的中文版本

生成AI智能校验的中文版本