多维级数模分圆多项式六次幂的 $q$-超同余式

A family of $q$-congruences modulo the square of a cyclotomic polynomial

Using Watson's terminating $_8ϕ_7$ transformation formula, we prove a family of $q$-congruences modulo the square of a cyclotomic polynomial, which were originally conjectured by the author and Zudilin [J. Math. Anal. Appl. 475 (2019), 1636--646]. As an application, we deduce two supercongruences modulo $p^4$ ($p$ is an odd prime) and their $q$-analogues. This also partially confirms a special case of Swisher's (H.3) conjecture.

Some $q$-congruences with parameters

Let $Φ_n(q)$ be the $n$-th cyclotomic polynomial in $q$. Recently, the author and Zudilin provide a creative microscoping method to prove some $q$-supercongruences mainly modulo $Φ_n(q)^3$ by introducing an additional parameter $a$. In this paper, we use this creative microscoping method to confirm some conjectures on $q$-supercongruences modulo $Φ_n(q)^2$. We also give some parameter-generalizations of known $q$-supercongruences. For instance, we present further generalizations of a $q$-analogue of a famous supercongruence of Rodriguez-Villegas: $$ \sum_{k=0}^{p-1}\frac{{2k\choose k}^2}{16^k}…

$q$-Supercongruences with parameters

In terms of the creative microscoping method recently introduced by Guo and Zudilin [Adv. Math. 346 (2019), 329--358], we find a $q$-supercongruence with four parameters modulo $Φ_n(q)(1-aq^n)(a-q^n)$, where $Φ_n(q)$ denotes the $n$-th cyclotomic polynomial in $q$. Then we empoly it and the Chinese remainder theorem for coprime polynomials to derive a $q$-supercongruence with two parameters modulo $[n]Φ_n(q)^3$, where $[n]=(1-q^n)/(1-q)$ is the $q$-integer.

来自 Watson 变换的两族 $q$-超同余式

Two Families of $q$-Supercongruences from Watson's Transformation

利用 Watson 的 ${}_{8}ϕ_7$ 变换、Guo 和 Zudilin 的创造性微观方法以及互素多项式的中国剩余定理，建立了模分圆多项式三次和四次幂的两族 $q$-超同余式。

多重基本超几何级数的若干$q$-超同余式

Some $q$-supercongruences for multiple basic hypergeometric series

该研究利用超几何级数公式、中国剩余定理等工具，建立了多重基本超几何级数在分圆多项式五、六次幂模下的$q$-超同余式，推广了已有结果并补充了新结论。

$q$-Supercongruences from Gasper and Rahman's summation formula

In 2017, He [Proc. Amer. Math. Soc. 145 (2017), 501--508] established two spuercongruences on truncated hypergeometric series and further proposed two related conjectures. Subsequently, Liu [Results Math. 72 (2017), 2057--2066] extended He's formulas and confirmed the second conjecture. However, the first conjecture is still open up to now. With the help of the creative microscoping method and the Chinese remainder theorem for coprime polynomials, we derive several $q$-supercongruences modulo the fourth and fifth powers of a cyclotomic polynomial from Gasper and Rahman's summation formula for …

两个由q级数求和公式导出的参数化q-超同余式

Two parametric q-supercongruences from a summation formula for q-series

本文利用q级数求和公式与创造性显微镜方法，建立了两个以分圆多项式三次幂为模的参数化q-超同余式，并在q→1时推导出一个具体的模p^3同余结论。