no code implementations • 4 Mar 2021 • Chen Wang, Zhi-Wei Sun
For any $m, n\in\mathbb{N}=\{0, 1, 2\ldots\}$, the truncated hypergeometric series ${}_{m+1}F_m$ is defined by $$ {}_{m+1}F_m\bigg[\begin{matrix}x_0&x_1&\ldots&x_m\\ &y_1&\ldots&y_m\end{matrix}\bigg|z\bigg]_n=\sum_{k=0}^n\frac{(x_0)_k(x_1)_k\cdots(x_m)_k}{(y_1)_k\cdots(y_m)_k}\cdot\frac{z^k}{k!
Number Theory Combinatorics 11A07, 33C20, 11B65, 05A10
no code implementations • 9 Dec 2020 • Hao Pan, Zhi-Wei Sun
For each $n=0, 1, 2,\ldots$ the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$.
Number Theory Combinatorics
no code implementations • 9 Sep 2020 • Zhi-Wei Sun
In this paper, we deduce a family of six new series for $1/\pi$; for example, $$\sum_{n=0}^\infty\frac{41673840n+4777111}{5780^n}W_n\left(\frac{1444}{1445}\right) =\frac{147758475}{\sqrt{95}\,\pi}$$ where $W_n(x)=\sum_{k=0}^n\binom nk\binom{n+k}k\binom{2k}k\binom{2(n-k)}{n-k}x^k$.
Number Theory Combinatorics 11B65, 05A19, 11A07, 11E25, 33F10
