### Electronic Research Archive

2020, Issue 3: 1273-1342. doi: 10.3934/era.2020070
Special Issues

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

• 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 $8$.

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

### Related Papers:

• 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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