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Some new series for $1/\pi $ motivated by congruences

Volume 173 / 2023

Zhi-Wei Sun Colloquium Mathematicum 173 (2023), 89-109 MSC: Primary 11B65; Secondary 05A19, 11A07, 11E25, 33F10. DOI: 10.4064/cm9017-2-2023 Published online: 3 April 2023


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 $W_n(x)=\sum_{k=0}^n\binom nk\binom{n+k}k\binom{2k}k\binom{2(n-k)}{n-k}x^k$. To do so, we manage to transform our series to series of the type $$\sum_{n=0}^\infty \frac{an+b}{m^n}\sum_{k=0}^n\binom nk^4$$ studied by Shaun Cooper in 2012. In addition, we pose $17$ new series for $1/\pi$ motivated by congruences; for example, we conjecture that $$\sum_{k=0}^\infty \frac{4290k+367}{3136^k}\binom{2k}kT_k(14,1)T_k(17,16)=\frac{5390}{\pi},$$ where $T_k(b,c)$ is the coefficient of $x^k$ in the expansion of $(x^2+bx+c)^k$.


  • Zhi-Wei SunDepartment of Mathematics
    Nanjing University
    Nanjing 210093, People’s Republic of China

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