论文标题
大约$ q $ - 符合某些身份
Some $q$-congruences arising from certain identities
论文作者
论文摘要
在本文中，通过构建某些身份，我们证明了一些一致性的$ Q $ - 纳瓦卢人。例如，对于任何奇数$ n> 1 $，我们表明\ begin {chater*} \ sum_ {k = 0}^{n-1} \ frac {（q^{ - 1}; q^2）_k} _k}} {（q; q; q; q; q; q; q; q} q^{（n^2-1）/4} - （1+q）[n] \ pmod {φ_n（q）^2}，\\ \ sum_ {k = 0}^{n-1} \ frac} \ frac {（q^3; q^3; q^2） q^{（n^2-9）/4} + \ frac {1 + q} {q^2} [n] \ pmod {φ_n（q）^2}，\ end {change*}，其中$ q $ q $ -pochhanmmer符号由$（x; x; x; x; x; q）_0 = 1 $和$（x $ and $ and q）定义（1-x）（1-xq）\ cdots（1-xq^{k-1}）$对于$ k \ geq1 $，$ q $ -integer由$ [n] = 1+q+q+q+\ cdots+q+q^{n-1} $和$φ_n（q）$ n $ n $ th cyclotomic polynomial nisthe上面的$ Q $ - 企业证实了Gu和Guo最近的一些猜想。
In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-1)/4} - (1+q)[n] \pmod{Φ_n(q)^2},\\ \sum_{k=0}^{n-1}\frac{(q^3;q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2} q^{(n^2-9)/4} + \frac{1+q}{q^2}[n]\pmod{Φ_n(q)^2}, \end{gather*} where the $q$-Pochhanmmer symbol is defined by $(x;q)_0=1$ and $(x;q)_k = (1-x)(1-xq)\cdots(1-xq^{k-1})$ for $k\geq1$, the $q$-integer is defined by $[n]=1+q+\cdots+q^{n-1}$ and $Φ_n(q)$ is the $n$-th cyclotomic polynomial. The $q$-congruences above confirm some recent conjectures of Gu and Guo.
