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

Zhi-Wei Sun; Zhi-Wei Sun

doi:10.3934/era.2020070

Electronic Research Archive

2020, Volume 28, Issue 3: 1273-1342. doi: 10.3934/era.2020070

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

Zhi-Wei Sun

Department of Mathematics, Nanjing University, Nanjing 210093, China

Received: 01 January 2020
Primary: 11A07, 11B65; Secondary: 05A19, 11E25, 33F10

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 $.
- Congruences,
- binomial coefficients,
- Ramanujan-type series for 1/π,
- binary quadratic forms,
- symbolic computation
Citation: Zhi-Wei Sun. New series for powers of $ \pi $ and related congruences[J]. Electronic Research Archive, 2020, 28(3): 1273-1342. doi: 10.3934/era.2020070

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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 $.

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