We build on and generalize recent work on so-termed factorization theorems for Lambert series generating functions. These factorization theorems allow us to express formal generating functions for special sums as invertible matrix transformations involving partition functions. In the Lambert series case, the generating functions at hand enumerate the divisor sum coefficients of \(q^n\) as \(\sum _{d|n} f(d)\) for some arithmetic function f. Our new factorization theorems provide analogs to these established expansions generating corresponding sums of the form \(\sum _{d: (d,n)=1} f(d)\) (type I sums) and the Anderson–Apostol sums \(\sum _{d|(m,n)} f(d) g(n/d)\) (type II sums) for any arithmetic functions f and g. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function p(n) to arbitrary arithmetic functions f. We conclude the last section of the article by directly expanding new formulas for an arithmetic function g by the type II sums using discrete, and discrete time, Fourier transforms (DFT and DTFT) for functions over inputs of greatest common divisors.

我们在Lambert级数生成函数的所谓因式分解定理的基础上，对最近的工作进行了总结和推广。这些分解定理使我们能够将特殊和的形式生成函数表示为涉及分区函数的可逆矩阵变换。在朗伯级数的情况下，对于某些算术函数f，生成函数将\（q ^ n \）的除数和系数枚举为\（\ sum _ {d | n} f（d）\）。我们新的分解定理为这些已建立的展开式提供了类似物，从而生成形式为\（\ sum _ {d：（d，n）= 1} f（d）\）（类型I的总和）和Anderson-Apostol的总和\ （\ sum _ {d |（m，n）} f（d）g（n / d）\）（任何算术函数的II型和）f和g。我们对II型总和的处理包括将分配函数p（n）与任意算术函数f相关的基于矩阵的分解方法。我们通过使用类型II的总和直接扩展算术函数g的新公式来结束本文的最后一部分，该函数使用离散和离散时间傅立叶变换（DFT和DTFT）对最大公约数的输入上的函数进行扩展。