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