论文标题
一致的ETA质量及其应用
Congruences for a class of eta-quotients and their applications
论文作者
论文摘要
可以使用生成函数来定义分区函数$ p _ {[1^c \ ell^d]}（n）$ \ [\ sum_ {n = 0}^{\ infty} p _ {[1^c {\ ell}^d]}（n）q^n = \ prod_ {n = 1}^{\ infty} {\ infty} {\ infty} \ dffrac {1} {1} {（1-q^n）{（1-q^n）^c（1-q^n）^c（1- n） \ cite {p}，我们证明了该分区功能的无限家族，以$ \ ell = 11 $。在本文中，我们扩展了我们在\ cite {p}中使用的想法，以证明无限的一致家庭的分区功能$ p _ {[1^c \ ell^d]}（n）$ modulo powers $ modulo powers $ \ ell $ of $ \ ell $的任何整数$ c $ and $ d $，用于primes $ 5 $，$ 5 $ $ $ \ el el el el el fle \ el \ elq 17 $ 17。这概括了Atkin，Gordon和Hughes的一致性分区功能。证明对一致性子组$γ_0（\ ell）$的模块化函数的向量空间进行了明确的基础。最后，我们使用这些一致性证明了广义的Frobenius $ \ ell $ -Color-color分区，$ \ ell- $常规分区和$ \ ell- $核心分区的一致性和不一致。
The partition function $ p_{[1^c\ell^d]}(n)$ can be defined using the generating function, \[\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}.\] In \cite{P}, we proved infinite family of congruences for this partition function for $\ell=11$. In this paper, we extend the ideas that we have used in \cite{P} to prove infinite families of congruences for the partition function $p_{[1^c\ell^d]}(n)$ modulo powers of $\ell$ for any integers $c$ and $d$, for primes $5\leq \ell\leq 17$. This generalizes Atkin, Gordon and Hughes' congruences for powers of the partition function. The proofs use an explicit basis for the vector space of modular functions of the congruence subgroup $Γ_0(\ell)$. Finally we used these congruences to prove congruences and incongruences of the generalized Frobenius $\ell$-color partitions, $\ell-$regular partitions and $\ell-$core partitions for $\ell=5,7,11,13$ and $17$.
