论文标题
分解定理和规范表示，用于生成特殊总和的函数
Factorization theorems and canonical representations for generating functions of special sums
论文作者
论文摘要
该手稿通过基于矩阵的某些类型的因素化来探讨许多卷积（限制的求和）类型序列，这些序列可用于表达其生成功能。论文的最后一个主要（非广告）部分探讨了如何最好地定义一个所谓的``\ emph {canonylyty thincy thincy}''基于给定的卷积总和序列的基于矩阵的分解。对于此类序列的生成函数的规范分解的概念需要匹配我们在Lambert系列生成函数（LGFS）的分解定理中发现的定性属性。我们在LGF案例中发现的预期最表现力的扩展自然而然地来自基础LGF系列类型的代数结构。我们提出了一个精确的定量要求，以根据某些序列定义了我们研究的生成函数扩展的基于矩阵的因素化的某些序列，以最佳的互相关统计量来概括此概念。最终，我们在能够获得给定总和类型的最大（最小）相关统计统计量时，就我们期望找到的矩阵因素化的类型提出了一些猜想。
This manuscript explores many convolution (restricted summation) type sequences via certain types of matrix based factorizations that can be used to express their generating functions. The last primary (non-appendix) section of the thesis explores the topic of how to best rigorously define a so-termed ``\emph{canonically best}'' matrix based factorization for a given class of convolution sum sequences. The notion of a canonical factorization for the generating function of such sequences needs to match the qualitative properties we find in the factorization theorems for Lambert series generating functions (LGFs). The expected qualitatively most expressive expansion we find in the LGF case results naturally from algebraic constructions of the underlying LGF series type. We propose a precise quantitative requirement to generalize this notion in terms of optimal cross-correlation statistics for certain sequences that define the matrix based factorizations of the generating function expansions we study. We finally pose a few conjectures on the types of matrix factorizations we expect to find when we are able to attain the maximal (respectively minimal) correlation statistic for a given sum type.
