Supercongruences on some binomial sums involving Lucas sequences

Abstract

In this paper, we confirm several conjectured congruences of Sun concerning the divisibility of binomial sums. For example, with help of a quadratic hypergeometric transformation, we prove that $$\sum _{k=0}^{p-1}\left(\genfrac{}{}{0ex}{}{p-1}{k}\right){\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\frac{P_k}{8^k}\equiv 0(\mathrm{mod}{p}^2)$$ for any prime $p\equiv 7\mathrm{mod}8$, where $P_k$ is the $k$-th Pell number. Further, we also propose three new congruences of the same type.

21 pages, to appear in Journal of Mathematical Analysis and Applications

References in corpus (3)

• Congruences concerning Legendre polynomials
• Some q-analogues of supercongruences of Rodriguez-Villegas
• Super congruences involving Bernoulli and Euler polynomials

Cited by in corpus (2)

• Proof of a conjecture of Adamchuk
• Three pairs of congruences concerning sums of central binomial coefficients