A family of polynomials and related congruences and series

Abstract

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum _{i,j=0}^n{\left(\genfrac{}{}{0ex}{}{n}{i}\right)}^m{\left(\genfrac{}{}{0ex}{}{n}{j}\right)}^m\left(\genfrac{}{}{0ex}{}{i+j}{i}\right)x^{i+j}(m,n=0,1,2,\dots ).$$ For example, we show that $$\sum _{k=0}^{p-1}{S}_k^{(0)}(x)\equiv \frac{x}{2x-1}\left(1+\left(\frac{1-4x^2}{p}\right)\right)(\mathrm{mod}p)$$ for any odd prime $p$ and integer $x\not\equiv 1/2(\mathrm{mod}p)$, where $(\frac{\cdot }{p})$ denotes the Legendre symbol. We also formulate some open conjectures on related congruences and series for $1/\pi$. For example, we conjecture that $$\sum _{k=0}^{\infty }(7k+1)\frac{S_k^{(2)}(1/11)}{9^k}=\frac{5445}{104\sqrt{39}\pi }$$ and $$\sum _{k=0}^{\infty }(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt{2}\pi }.$$

arXiv comment23 pages, final published version