Factorization theorems and canonical representations for generating functions of special sums

Schmidt, Maxie Dion

Title:

Factorization theorems and canonical representations for generating functions of special sums

dc.contributor.advisor	Yu, Josephine
dc.contributor.author	Schmidt, Maxie Dion
dc.contributor.committeeMember	Baker, Matthew
dc.contributor.committeeMember	de la Llave, Rafael
dc.contributor.committeeMember	Athreya, Jayadev
dc.contributor.committeeMember	Berndt, Bruce
dc.contributor.department	Mathematics
dc.date.accessioned	2022-08-25T13:35:29Z
dc.date.available	2022-08-25T13:35:29Z
dc.date.created	2022-08
dc.date.issued	2022-07-19
dc.date.submitted	August 2022
dc.date.updated	2022-08-25T13:35:29Z
dc.description.abstract	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 "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.
dc.description.degree	Ph.D.
dc.format.mimetype	application/pdf
dc.identifier.uri	http://hdl.handle.net/1853/67248
dc.language.iso	en_US
dc.publisher	Georgia Institute of Technology
dc.subject	generating function
dc.subject	convolution type sum
dc.subject	divisor sum
dc.subject	Lambert series
dc.subject	partition function.
dc.title	Factorization theorems and canonical representations for generating functions of special sums
dc.type	Text
dc.type.genre	Dissertation
dspace.entity.type	Publication
local.contributor.advisor	Yu, Josephine
local.contributor.corporatename	College of Sciences
local.contributor.corporatename	School of Mathematics
relation.isAdvisorOfPublication	0569eb14-6836-43ce-bbd4-195f87a50534
relation.isOrgUnitOfPublication	85042be6-2d68-4e07-b384-e1f908fae48a
relation.isOrgUnitOfPublication	84e5d930-8c17-4e24-96cc-63f5ab63da69
thesis.degree.level	Doctoral