Deformation and Symmetry in Two Hypergeometric Supercongruences
Abstract
We prove two supercongruence conjectures of Li--Long--Tu concerning truncated very-well-poised 7F6(1) series and Fourier coefficients of modular forms. The proofs are based on a common deformation method. By deforming the original truncated hypergeometric series to suitable terminating series, we are able to apply classical hypergeometric transformations together with Mishev's W(D5) symmetry. This reduces both problems to congruences for lower-rank hypergeometric sums that can be treated using the p-adic methods of Long--Tu--Yui--Zudilin. Although the two cases require different local arithmetic calculations, their global proof structure is the same. The resulting argument gives a unified explanation of these two supercongruences and suggests that the same deformation-and-symmetry method may apply more broadly.
Related
- Coxeter Symmetry in Hypergeometric Functions and Elliptic Integral Moments
- q-Analogues of some supercongruences related to Van Hamme's (E.2) and (F.2) supercongruences
- q-Analogues of some supercongruences related to generalized Van Hamme-type supercongruences
- q-Supercongruences Modulo the Third and Fifth Powers of a Cyclotomic Polynomial
- Proof of Two Conjectural q-Supercongruences Related to Van Hamme's (G.2) Supercongruence
BibTeX
@misc{bao2026deformation,
title = {{Deformation and Symmetry in Two Hypergeometric Supercongruences}},
author = {Dianbin Bao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15176},
eprint = {2610.15176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15176},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Deformation and Symmetry in Two Hypergeometric Supercongruences
AU - Dianbin Bao
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.15176
UR - https://arxiv.org/abs/2610.15176
AB - We prove two supercongruence conjectures of Li--Long--Tu concerning truncated very-well-poised _7F_6(1) series and Fourier coefficients of modular forms. The proofs are based on a common deformation method. By deforming the original truncated hypergeometric series to suitable terminating series, we are able to apply classical hypergeometric transformations together with Mishev's W(D_5) symmetry. This reduces both problems to congruences for lower-rank hypergeometric sums that can be treated using the p-adic methods of Long--Tu--Yui--Zudilin. Although the two cases require different local arithmetic calculations, their global proof structure is the same. The resulting argument gives a unified explanation of these two supercongruences and suggests that the same deformation-and-symmetry method may apply more broadly.
ER -