Finite-sum Identities and Supercongruences for Cubes of Catalan Numbers

Abstract

Let $C_k=\binom{2k}{k}/(k+1)$ be the $k$-th Catalan number, and let $p$ be an odd prime. We establish exact finite-sum identities that, together with known binomial supercongruences and reductions of Sun, yield $\sum_{k=0}^{(p-1)/2}\frac{C_k^3}{16^k}\equiv2-24y^2-\frac{5p^2}{2y^2}\pmod{p^3}$ when $p=x^2+3y^2\equiv1\pmod3$, and $\sum_{k=0}^{(p-1)/2}\frac{C_k^3}{(-8)^k}\equiv-1-48y^2-\frac{33p^2}{16y^2}\pmod{p^3}$ when $p=x^2+4y^2\equiv1\pmod4$. We also evaluate the corresponding sum with denominator $16^k(k+1)^2$, and obtain two weighted Catalan supercongruences. The finite identities follow from the ratios of consecutive summands, symmetry of double sums, and telescoping. All endpoint terms are retained over the rational numbers before any modular reduction. In each bilinear identity, one endpoint denominator equals $p$; its cancellation produces the necessary term of order $p^2$. The precise arithmetic inputs are stated separately.