A family of polynomials and related congruences and series

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum _{i,j=0}^n{\left(\genfrac{}{}{0px}{}{n}{i}\right)}^m{\left(\genfrac{}{}{0px}{}{n}{j}\right)}^m\left(\genfrac{}{}{0px}{}{i+j}{i}\right)x^{i+j}(m,n=0,1,2,\dots )\mathrm{.}$$ For example, we show that $$\sum _{k=0}^{p-1}{S}_k^{(0)}(x)\equiv \frac{x}{2x-1}(1+\left(\frac{1-4x^2}{p}\right))(\mathrm{m}\mathrm{o}\mathrm{d}p)$$ for any odd prime $p$ and integer $x\equiv ̸1\mathrm{/}2(\mathrm{m}\mathrm{o}\mathrm{d}p)$, where $(\frac{\cdot }{p})$ denotes the Legendre symbol. We also formulate some open conjectures on related congruences and series for $1\mathrm{/}\pi$. For example, we conjecture that $$\sum _{k=0}^{\mathrm{\infty }}(7k+1)\frac{S_k^{(2)}(1\mathrm{/}11)}{9^k}=\frac{5445}{104\sqrt{39}\pi }$$ and $$\sum _{k=0}^{\mathrm{\infty }}(1365k+181)\frac{S_k^{(2)}(1\mathrm{/}18)}{{16}^k}=\frac{1377}{\sqrt{2}\pi }\mathrm{.}$$