Congruences concerning binomial coefficients and binary quadratic forms

TL;DR

This paper derives congruences modulo p^2 for specific binomial coefficient sums involving quadratic forms and weights, extending known results and providing partial results for related sums with various parameters.

Contribution

It introduces new congruences for binomial sums involving quadratic forms and weights modulo p^2, expanding the understanding of binomial coefficient congruences.

Findings

Derived congruences for sums involving binomial coefficients and quadratic forms modulo p^2.

Provided partial results for sums with different parameters and weights.

Extended previous work on binomial coefficient congruences in number theory.

Abstract

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