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Weighted bilinear identities and supercongruences for Apéry-like polynomials
Abstract
We establish weighted bilinear summation identities for two families of Apéry-like polynomials gn(x) and vn(x). The identities express weighted sums in terms of consecutive endpoint values and, when necessary, lower moments. For gn(x)2 we obtain identities with weights (2n+1)r for 1≤ r≤4; for vn(x)2 we treat the cubic and quintic weights. Combining these formulas with congruences for the endpoint values gives supercongruences modulo p3 and p4, together with special evaluations modulo p5 and p7, where p is a prime greater than $3$. In particular, ∑n=0p−1(2n+1)3vn (5/2)2 ≡ 6p4−143/3p6pmod p7, confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and p-adic endpoint expansions.
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BibTeX
@misc{li2026weighted,
title = {{Weighted bilinear identities and supercongruences for Apéry-like polynomials}},
author = {Yu-Tian Li and Zhi-Hong Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28098},
eprint = {2609.28098},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28098},
}
RIS
TY - UNPB
TI - Weighted bilinear identities and supercongruences for Apéry-like polynomials
AU - Yu-Tian Li
AU - Zhi-Hong Sun
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28098
UR - https://arxiv.org/abs/2609.28098
AB - 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≤ r≤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, ∑_n=0^p−1(2n+1)^3v_n (5/2)^2 ≡ 6p^4−143/3p^6pmod p^7, confirming a congruence conjectured by Sun. The proofs use explicit quadratic telescoping identities and p-adic endpoint expansions.
ER -