New series for powers of $ \pi $ and related congruences

Abstract

Via symbolic computation we deduce 97 new type series for powers of $ \pi $ related to Ramanujan-type series. Here are three typical examples: $ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $ with $ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $ $ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $ and $ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $ where the generalized central trinomial coefficient $ T_k(b,c) $ denotes the coefficient of $ x^k $ in the expansion of $ (x^2+bx+c)^k $. We also formulate a general characterization of rational Ramanujan-type series for $ 1/\pi $ via congruences, and pose 117 new conjectural series for powers of $ \pi $ via looking for corresponding congruences. For example, we conjecture that $ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $ Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $ 8 $. Via symbolic computation we deduce 97 new type series for powers of $ \pi $ related to Ramanujan-type series. Here are three typical examples: $ \sum\limits_{k = 0}^\infty\frac{P(k)\binom{2k}k\binom{3k}k\binom{6k}{3k}}{(k+1)(2k-1)(6k-1)(-640320)^{3k}} = \frac{18\times557403^3\sqrt{10005}}{5\pi} $ with $ \begin{align*} P(k) = &637379600041024803108 k^2 + 657229991696087780968 k \\&+ 19850391655004126179, \end{align*} $ $ \sum\limits_{k = 1}^\infty \frac{(3k+1)16^k}{(2k+1)^2k^3 \binom{2k}k^3} = \frac{\pi^2-8}2, $ and $ \sum\limits_{n = 0}^\infty\frac{3n+1}{(-100)^n}\sum\limits_{k = 0}^n{n\choose k}^2T_k(1,25)T_{n-k}(1,25) = \frac{25}{8\pi}, $ where the generalized central trinomial coefficient $ T_k(b,c) $ denotes the coefficient of $ x^k $ in the expansion of $ (x^2+bx+c)^k $. We also formulate a general characterization of rational Ramanujan-type series for $ 1/\pi $ via congruences, and pose 117 new conjectural series for powers of $ \pi $ via looking for corresponding congruences. For example, we conjecture that $ \sum\limits_{k = 0}^\infty\frac{39480k+7321}{(-29700)^k}T_k(14,1)T_k(11,-11)^2 = \frac{6795\sqrt5}{\pi}. $ Eighteen of the new series in this paper involve some imaginary quadratic fields with class number $ 8 $.