A family of polynomials and related congruences and series

Abstract

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac x{2x-1}\left(1+\left(\frac{1-4x^2}p\right)\right)\pmod p $$ for any odd prime $p$ and integer $x\not\equiv1/2\pmod p$, 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 $$\sum_{k=0}^\infty(1365k+181)\frac{S_k^{(2)}(1/18)}{16^k}=\frac{1377}{\sqrt2\,\pi}.$$