Sums of cubes of Delannoy polynomials modulo p³

Let D_k(x) = Σ_j C(k,j) C(k+j,j) x^j be the Delannoy polynomials (D_k(x) = P_k(2x+1), with P_k the Legendre polynomials) and T_p(z) = Σk<p C(2k,k) z^k. Z.-W. Sun conjectured (Sci. China Math. 57 (2014), Conjecture 5.1), and V. J. W. Guo proved (Integral Transforms Spec. Funct. 26 (2015)), that Σₖ₌₀ᵖ⁻¹ (2k+1) D_k(x)³ ≡ p ((−4x−3)/p) (mod p²) for every prime p not dividing x(x+1). Result. We determine this sum modulo p³. For every odd prime p not dividing x, Σₖ₌₀ᵖ⁻¹ (2k+1) D_k(x)³ ≡ p χ + (3(2x+1)/(2x)) p (T_p(x+1) − χ) (mod p³), where χ = ((−4x−3)/p). Together with Sun's evaluation of T_p modulo p² (Sci. China Math. 53 (2010)), this gives, for p not dividing x(x+1)(4x+3), Σₖ₌₀ᵖ⁻¹ (2k+1) D_k(x)³ ≡ p χ + (3(2x+1)/x) p W_p (mod p³), where u_n is the Lucas sequence u_0 = 0, u_1 = 1, uₙ₊₁ = u_n − (x+1) un−1, and W_p = (x+1) up−1 if χ = 1, W_p = uₚ₊₁ if χ = −1 (p divides W_p, so W_p/p is a Lucas quotient). Replacing x by −1−x gives the alternating sums Σ (−1)^k (2k+1) D_k(x)³ modulo p³, which refines a second congruence of Guo. For the central Delannoy numbers D_k = D_k(1), Σk<p (−1)^k (2k+1) D_k³ ≡ p (p/5) (1 − (9/2) Fp−(p/5)) (mod p³) for p ≠ 2, 5, where F_n are the Fibonacci numbers. The congruence does not hold modulo p⁴ in general. Discovery. The congruence was discovered experimentally with SyntheticMind, the mathematical discovery engine created by the author. Its residue laboratory computed (S_p(x) − pχ)/p² modulo p prime by prime. Its fitting laboratory (lattice reduction, with separate control primes) found no relation with Fermat quotients or the Bernoulli and Euler numbers Bp−3, Ep−3. It found the exact law once the Lucas quotient up−χ(1, x+1)/p was added. Method. Laplace's product formula for Legendre polynomials gives D_k(x)² = Λ[P_k(1+2X)] for the linear functional Λ(X^j) = C(2j,j)(x²+x)^j. The Christoffel–Darboux formula then turns the whole sum into a difference quotient of P_p and Pp−1. This gives a short new proof of the Sun–Guo congruence. It also reduces the p³ statement to two ingredients: Sun's Lucas-quotient theorem, and the evaluation modulo p of a double sum with harmonic weights. That evaluation is done with an integral representation, a Glaisher-type congruence and the Binet formulas. Verification. Every lemma and intermediate congruence was checked with exact modular arithmetic, with no failures. The main theorem was checked on 4188 pairs (x, p) with |x| ≤ 60 and 3 ≤ p < 160, including the cases p | x+1 and p | 4x+3. The Lucas form and each step of the proof were checked on 1868 further pairs with p < 300, and both corollaries for the central Delannoy numbers for p < 800. The paper was also examined by an independent referee pass (a separate instance of the same AI system), which found no mathematical error. Use of AI. The laboratories of SyntheticMind used here were programmed with the assistance of the AI system Claude (Anthropic; Claude Opus 5.5) under the author's direction. The proofs and the text were also developed with its assistance, as described in Section 2.3 of the paper. The author takes full responsibility for the content. Files: Blanco_Gomez_2026_Delannoy_cubes_p3_EN.pdf: the paper. Blanco_Gomez_2026_Delannoy_cubes_p3_ES.pdf: the Spanish version, with the same content. verify_delannoy_cubes_p3.py: the verification script (Python 3, standard library only). It checks every statement and every intermediate step of the proofs, runs in a few seconds and prints ALL CHECKS PASSED. LOG_VERIFICACION.txt: the output of the script. sources_tex.zip: the LaTeX sources.