Congruences concerning binomial coefficients and binary quadratic forms

Abstract

Let $p>3$ be a prime. In this paper, we obtain the congruences for $$\sum _{k=0}^{p-1}\frac{w(k){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3}{(-8{)}^k},\sum _{k=0}^{p-1}\frac{w(k){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{3k}{k}\right)}{(-192{)}^k},\sum _{k=0}^{p-1}\frac{w(k){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)}{(-144{)}^k}\text{and}\sum _{k=0}^{p-1}\frac{w(k){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{4k}{2k}\right)}{648^k}$$ modulo $p^2$, and partial results for ${\sum }_{k=0}^{(p-1)/2}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^3\frac{w(k)}{m^k}$ modulo $p^2$, where $m\in \{1,16,-64,256,-512,4096\}$ and $w(k)\in \{k^2,k^3,\frac{1}{k+1},\frac{1}{(k+1{)}^2},\frac{1}{(k+1{)}^3},\frac{1}{2k-1},\frac{1}{k+2}\}$.

correct some typos