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Nazaroglu, Caner ORCID: 0000-0001-8844-2441, Pandey, Badri Vishal ORCID: 0000-0002-8226-3028 and Singh, Ajit (2025). Quasimodularity and limiting behavior for variations of MacMahon series. Advances in Mathematics, 482. pp. 1-26. Elsevier. ISSN 0001-8708

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Abstract

[Artikel-Nr.: 110565] Motivated by the 1920’s seminal work of Major MacMahon, Amdeberhan–Andrews–Tauraso recently introduced an infinite family of q-series �t(a; q) :=∑︂ 1≤n1<n2<···<nt qn1 +n2 +···+nt (1+aqn1 +q2n1)(1+aqn2 +q2n2 )···(1+aqnt +q2nt) and proved that these functions are linear combinations of quasimodular forms. In this paper, we study a broader family of q-series that contains the collection {�t}t∈N. Using the theory of quasi shuffle algebras, we show that this extended family also lies in the algebra of quasimodular forms. More over, we determine the precise weights and levels of these functions, thereby making Amdeberhan–Andrews–Tauraso’s result sharp. We further investigate the limiting behavior of these functions. In particular, we demonstrate that the sequence of quasimodular forms {�t(1; q)}t∈N gives an approximation for the ordinary partition function. We also establish infinitely many closed formulas for reciprocals of certain infinite products in terms �t(a; q).

Item Type:	Article
Creators:	Creators Email ORCID ORCID Put Code Nazaroglu, Caner UNSPECIFIED https://orcid.org/0000-0001-8844-2441 UNSPECIFIED Pandey, Badri Vishal UNSPECIFIED https://orcid.org/0000-0002-8226-3028 UNSPECIFIED Singh, Ajit UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-810128
Identification Number:	10.1016/j.aim.2025.110565
Journal or Publication Title:	Advances in Mathematics
Volume:	482
Page Range:	pp. 1-26
Number of Pages:	26
Date:	December 2025
Publisher:	Elsevier
ISSN:	0001-8708
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 Eisenstein series ; Quasimodular forms ; Quasi-shuffle algebra ; MacMahon-type q-series English
['eprint_fieldname_oa_funders' not defined]:	Publikationsfonds UzK
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/81012