论文标题
关于广义的Frobenius分区功能$ ϕ_K（n）$的均等
On the Parity of the Generalized Frobenius Partition Functions $ϕ_k(n)$
论文作者
论文摘要
乔治·安德鲁斯（George Andrews）在1984年的《美国数学学会回忆录》中定义了两个功能系列，分别是$ ϕ_k（n）$和$ cx_k（n），$列举了两种类型的组合物体，而安德鲁斯称为概括性的frobenius分区。作为回忆录的一部分，安德鲁斯证明了这两个家庭中的特定功能所满足的许多拉曼努扬（Ramanujan）。在随后的几年中，许多其他作者证明了这些功能的相似结果，通常是为了在此简短说明中特定选择$k。$的特定选择，我们的目标是确定$ k $的{\ bf infinite} novely $ k $的norlage n $ n $ ϕ_k（n）$，即使在特定的$ n $中，在特定的$ n $中都适用特别是，我们在这项工作中的主要目标是证明，对于所有积极整数$ \ ell，$ all primes $ p \ geq 5，$ and $ r，$ $ 0 <r <p，$ $ 24r+1 $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ P，$ P，$ $ P，$ $$ ϕ_ {p \ ell-1}（pn+r）\ equiv 0 \ pmod {2} 所有$ n \ geq 0的$$。$ 我们对此结果的证明是真正的基础，依赖于安德鲁斯回忆录，古典$ Q $的引理 - 系列结果和基本生成功能操作。这种结果适用于无限的$ k值，$很少在概括的Frobenius分区所满足的算术属性中，这主要是因为所讨论的生成函数的笨拙性质。
In his 1984 Memoir of the American Mathematical Society, George Andrews defined two families of functions, $ϕ_k(n)$ and $cϕ_k(n),$ which enumerate two types of combinatorial objects which Andrews called generalized Frobenius partitions. As part of that Memoir, Andrews proved a number of Ramanujan--like congruences satisfied by specific functions within these two families. In the years that followed, numerous other authors proved similar results for these functions, often with a view towards a specific choice of the parameter $k.$ In this brief note, our goal is to identify an {\bf infinite} family of values of $k$ such that $ϕ_k(n)$ is even for all $n$ in a specific arithmetic progression; in particular, our primary goal in this work is to prove that, for all positive integers $\ell,$ all primes $p\geq 5,$ and all values $r,$ $0 < r < p,$ such that $24r+1$ is a quadratic nonresidue modulo $p,$ $$ ϕ_{p\ell-1}(pn+r) \equiv 0 \pmod{2} $$ for all $n\geq 0.$ Our proof of this result is truly elementary, relying on a lemma from Andrews' Memoir, classical $q$--series results, and elementary generating function manipulations. Such a result, which holds for infinitely many values of $k,$ is rare in the study of arithmetic properties satisfied by generalized Frobenius partitions, primarily because of the unwieldy nature of the generating functions in question.
