Let \(p_k(n)\) be given by the series expansion of the k-th power of the Euler Product \(\prod _{n=1}^{\infty }(1-q^n)^k=\sum _{n=0}^{\infty }p_k(n)q^{n}\). By investigating the properties of the modular equations of the second and the third order under the Atkin U-operator, we determine the generating functions of \(p_{8k}(2^{2\alpha } n +\frac{k(2^{2\alpha }-1)}{3})\)\((1\le k\le 3)\) and \(p_{3k}(3^{2\beta }n+\frac{k(3^{2\beta }-1)}{8})\)\((1\le k\le 8)\) in terms of some linear recurring sequences. Combining with a result of Engstrom about the periodicity of linear recurring sequences modulo m, we obtain infinite families of congruences for \(p_k(n)\) modulo any \(m\ge 2\), where \(1\le k\le 24\) and 3|k or 8|k. Based on these congruences for \(p_k(n)\), infinite families of congruences for many partition functions such as the overpartition function, t-core partition functions and \(\ell \)-regular partition functions are easily obtained.

中文翻译：

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欧拉积的幂系数的同余

令\（p_k（n）\）由欧拉积的k次幂的级数展开给出\（\ prod _ {n = 1} ^ {\ infty}（1-q ^ n）^ k = \ sum _ {n = 0} ^ {\ infty} p_k（n）q ^ {n} \）。通过研究Atkin U算子下二阶和三阶模方程的性质，我们确定\（p_ {8k}（2 ^ {2 \ alpha} n + \ frac {k（2 ^ {2 \ alpha -1）} {3}）\）\（（1 \ le k \ le 3）\）和\（p_ {3k}（3 ^ {2 \ beta} n + \ frac {k（ 3 ^ {2 \ beta} -1）} {8}）\）\（（1 \ le k \ le 8）\）在某些线性重复序列方面。与Engstrom的结果有关以m为模的线性循环序列的周期，我们得到\（p_k（n）\）的无穷余等价族，取任何\（m \ ge 2 \）的模，其中\（1 \ le k \ le 24 \）和3 | k或8 | ķ。基于\（p_k（n）\）的这些等价关系，可以轻松获得许多分区函数（如过度分配函数，t核分区函数和\（\ ell \） -常规分区函数）的无限等价族。