Congruences concerning binomial coefficients and binary quadratic forms

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

Congruences concerning binomial coefficients and binary quadratic forms

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Congruences concerning binomial coefficients and binary quadratic forms

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Congruences concerning binomial coefficients and binary quadratic forms

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Congruences concerning binomial coefficients and binary quadratic forms

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