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

Abstract

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

20 pages.Accepted version for publication in Colloq. Math

References in corpus (1)

• Open Conjectures on Congruences