A Dwork-type congruence for Delannoy polynomials: proof of congruence (4.13) in Z.-W. Sun's Conjecture 63

Let D_n(x) = Σ_k C(n,k) C(n+k,k) x^k be the Delannoy polynomials, so that D_n(1) are the central Delannoy numbers (OEIS A001850: 1, 3, 13, 63, 321, …) and D_n(x) = P_n(2x+1) in terms of the Legendre polynomials. This note proves that for every prime p (p = 2 included), every integer n ≥ 1 and every p-adic integer x with x ≢ 0 (mod p), (1/n²) · ( Σₖ₌₀ᵖⁿ⁻¹ (2k+1) D_k(x) − p · Σₖ₌₀ⁿ⁻¹ (2k+1) D_k(x) ) ≡ 0 (mod p²). This is congruence (4.13) of Conjecture 63 in Z.-W. Sun's list "Open conjectures on congruences" (Nanjing Univ. J. Math. Biquarterly 36, 2019, p. 56), where it appears as open; the case n = 1 was proved by Sun for odd p (Sci. China Math. 57, 2014). Equivalently, T(mp^r) ≡ p·T(mp^(r−1)) (mod p^(2r)) for all m, r ≥ 1, where T(N) = Σk<N (2k+1) D_k(x). The hypothesis x ≢ 0 (mod p) cannot be dropped, and the exponent cannot be improved in general. How hard is it — two proofs. By a formula of Sun, T(N) = N·F_N(x) with F_N(x) = Σ_k C(N,k+1) C(N+k,k) x^k, so the statement is equivalent to the Gauss congruences Fₚₙ(x) ≡ F_n(x) (mod p^(1+ν_p(n))). For odd p these follow in a few lines from the classical Gauss congruences for sequences of the form [y^N] H(y)^N, and the note gives this short argument; for x = 1 (OEIS A002002) the Gauss congruences were already recorded in the OEIS. The result is therefore modest. The main argument of the note is different and covers every prime, including p = 2, where the short argument loses a factor 2: a transfer principle for the Legendre transform L(a)_n = Σ_k C(n,k) C(n+k,k) a_k, proved here at "level one" and for every prime. If the alternating partial sums of a p-integral sequence a satisfy A(mp^r) ≡ χ·A(mp^(r−1)) (mod p^r), then the (2k+1)-weighted alternating partial sums of L(a) satisfy the same tower modulo p^(2r) with constant χ·p. (The level-two version, odd primes, is in the note on Domb numbers, doi:10.5281/zenodo.23762821.) For the Delannoy polynomials the relevant sequence is geometric, a_k = (−1−x)^k, and its tower follows from Euler's theorem. What remains open. Transfer principles reduce the other three congruences of Conjecture 63, (4.14)–(4.16), for the squares, cubes and fourth powers of D_k(x), to towers for other sums. For (4.14), the relevant sum is that of c_k = Σ_j C(k,j) C(2j,j) (x²+x)^j, a family of generalized central trinomial coefficients. For (4.15) and (4.16), the relevant sequences have no known closed form. These towers hold numerically with exactly Sun's constants, but they are not proved; (4.14)–(4.16) remain open. Verification. The script verify_delannoy.py (Python 3, standard library only, about 20 seconds; 18615 checks, no failures) checks in exact rational arithmetic the identities used, the representations and Gauss congruences of the short proof (also for p = 2), the lemmas of the transfer principle for p ∈ {2, 3, 5, 7, 11}, and the theorem for every prime p ≤ 13 (p = 2 included), 13 values of x and every n with pn < 400 (5086 congruences); it also checks, numerically, the reductions of (4.14)–(4.16). The proofs do not depend on these computations. Authorship. The author of this work is Roberto Blanco Gómez. The main result and the transfer-principle approach were obtained with SyntheticMind, the mathematical reasoning engine that the author created, designed and programmed himself: it was found during the development of its version 50, a "corollary harvester" that combines the transfer principle with Legendre-transform identities and with Sun's list, and its "inverse reduction" module produced the reductions of (4.14)–(4.16). The AI system Claude (Anthropic) collaborated as an assistant, under the author's direction, in the formulation and checking of the proofs, the literature search, the verification script and the writing; a separate Claude session acting as an independent referee pointed out the short proof for odd p and the OEIS entry A002002. The author has checked the arguments and takes full responsibility for the content. Files: Blanco_Gomez_2026_Delannoy_Sun63_EN.pdf: the note (English). Blanco_Gomez_2026_Delannoy_Sun63_ES.pdf: the Spanish version, with the same content. verify_delannoy.py: the verification script (Python 3, standard library only). verify_delannoy_log.txt: its output. sources_tex.zip: the LaTeX sources.