تقرير
Supercongruences via Beukers' method
العنوان: | Supercongruences via Beukers' method |
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المؤلفون: | Sun, Zhi-Hong, Ye, Dongxi |
سنة النشر: | 2024 |
المجموعة: | Mathematics |
مصطلحات موضوعية: | Mathematics - Number Theory, Mathematics - Classical Analysis and ODEs, Mathematics - Combinatorics, 11A07, 11B65, 11E25, 11F03, 11F20 |
الوصف: | Recently, using modular forms F. Beukers posed a unified method that can deal with a large number of supercongruences involving binomial coefficients and Ap\'ery-like numbers. In this paper, we use Beukers' method to prove some conjectures of the first author concerning the congruences for $$\sum_{k=0}^{(p-1)/2}\frac{\binom{2k}k^3}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{m^k}, \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k\binom{6k}{3k}}{m^k}, \ \sum_{n=0}^{p-1}\frac{V_n}{m^n},\ \sum_{n=0}^{p-1}\frac{T_n}{m^n},\ \sum_{n=0}^{p-1}\frac{D_n}{m^n} $$ and $\sum_{n=0}^{p-1}(-1)^nA_n$ modulo $p^3$, where $p$ is an odd prime representable by some suitable binary quadratic form, $m$ is an integer not divisible by $p$, $V_n=\sum_{k=0}^n\binom{2k}k^2\binom{2n-2k}{n-k}^2$, $T_n=\sum_{k=0}^n\binom nk^2\binom{2k}n^2$, $D_n=\sum_{k=0}^n\binom nk^2\binom{2k}k\binom{2n-2k}{n-k}$ and $A_n$ is the Ap\'ery number given by $A_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k^2$. Comment: add more congruences |
نوع الوثيقة: | Working Paper |
URL الوصول: | http://arxiv.org/abs/2408.09776 |
رقم الأكسشن: | edsarx.2408.09776 |
قاعدة البيانات: | arXiv |
الوصف غير متاح. |