Universität zu Köln

Garnowski, Taylor (2022). Asymptotic Analysis of Mixed Mock Modular Forms and Related q-products. PhD thesis, Universität zu Köln.

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Abstract

This thesis contains results of three research projects which study asymptotics for the Fourier coefficients of mixed mock modular forms and twisted q-products arising in combinatorics. To begin, we compute an asymptotic distribution for generalizations of unimodal sequences called odd-balanced unimodal sequences which were de- fined by Kim, Lim, and Lovejoy in 2016. We find the interesting result that the odd-balanced unimodal sequences with certain restric- tions on their rank, are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which are asymptotically related to the partition function. In the second part of this thesis, we compute asymptotic estimates for the Fourier coefficients of two mock theta functions originating from Bailey pairs derived by Lovejoy and Osburn in 2012. We encounter cancellation in our estimates for one of the functions, which requires a careful study of secondary asymptotic terms. We deal with this by using higher order asymptotic expansions for the Jacobi theta functions. In our final result, we find asymptotic estimates for the complex Fourier coefficients of the product (ζq;q)−1, with ζ a root of unity. This result has interesting applications in analysis and combinatorics. For large n, we are able to predict sign changes of arbitrary linear combinations of the function p(a,b;n) for fixed b, where p(a,b;n) counts the number of partitions of n where the number of parts is congruent to a modulo b. We see that simple differences of the type p(a1 , b; n) − p(a2 , b; n) have sign change patterns that oscillate like a cosine.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Garnowski, Taylor UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-618951
Date:	30 May 2022
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Modular Forms UNSPECIFIED Circle Method UNSPECIFIED Partitions UNSPECIFIED
Date of oral exam:	30 May 2022
Referee:	Name Academic Title Bringmann, Kathrin Professor
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/61895