Weighted bilinear identities and supercongruences for Apéry-like polynomials

Abstract

We establish weighted bilinear summation identities for two families of Apéry-like polynomials $g_n(x)$ and $v_n(x)$. The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For $g_n(x{)}^2$ we obtain identities with weights $(2n+1{)}^r$ for $1\le r\le 4$; for $v_n(x{)}^2$ we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo $p^3$ and $p^4$, together with special evaluations modulo $p^5$ and $p^7$, where $p$ is a prime greater than $3$. In particular, $$\sum _{n=0}^{p-1}(2n+1{)}^3{v}_n{\left(\frac{5}{2}\right)}^2\equiv 6p^4-\frac{143}{3}{p}^6(\mathrm{mod}{p}^7),$$ confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and $p$-adic endpoint expansions.

29 pages