分区-频率枚举矩阵
The Partition-Frequency Enumeration Matrix
AI总结：
本文提出分区-频率枚举矩阵，通过无限上三角数论矩阵提供初等方法，统一数论函数与分区函数关系，并恢复扩展多个经典结果，如除数函数和拉马努金τ函数的递推关系。
摘要：
我们开发了一种算术，提供了一种初等方法，通过一个无限上三角数论矩阵来枚举类似分区的对象。我们称这个矩阵为分区-频率枚举（PFE）矩阵。这个矩阵统一了大量将数论函数与分区类型函数联系起来的结果。该算术被扩展到任意生成函数和具有Weierstrass乘积的函数。作为副产品，我们恢复（并扩展）了一些广为人知的递推关系，包括除数函数、拉马努金的τ函数、平方和三角数和的求和，以及ζ(2n)，其中n是正整数。这些包括欧拉、埃文、拉马努金、勒默等人的经典结果。作为一项应用，我们将拉马努金著名的同余式p(5n+4)≡0 (mod 5)和τ(5n+5)≡0 (mod 5)嵌入到无限族这样的同余式中。
英文摘要：
We develop a calculus that gives an elementary approach to enumerate partition-like objects using an infinite upper-triangular number-theoretic matrix. We call this matrix the Partition-Frequency Enumeration (PFE) matrix. This matrix unifies a large number of results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions, and functions with Weierstrass products. As a by-product, we recover (and extend) some well-known recurrence relations for many number-theoretic functions, including the sum of divisors function, Ramanujan's $τ$ function, sums of squares and triangular numbers, and for $ζ(2n)$, where $n$ is a positive integer. These include classical results due to Euler, Ewell, Ramanujan, Lehmer and others. As one application, we embed Ramanujan's famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $τ(5n+5)\equiv 0$ (mod $5)$ into an infinite family of such congruences.
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