Titel
On the infinite Borwein product raised to a positive real power
Nian Hong Zhou
School of Mathematics and Statistics, Guangxi Normal University
Abstract
In this paper, we study properties of the coefficients appearing in the q-series expansion of ∏n≥1[(1−qn)/(1−qpn)]δ, the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power δ. We use the Hardy–Ramanujan–Rademacher circle method to give an asymptotic formula for the coefficients. For p=3 we give an estimate of their growth which enables us to partially confirm an earlier conjecture of the first author concerning an observed sign pattern of the coefficients when the exponent δ is within a specified range of positive real numbers. We further establish some vanishing and divisibility properties of the coefficients of the cube of the infinite Borwein product. We conclude with an Appendix presenting several new conjectures on precise sign patterns of infinite products raised to a real power which are similar to the conjecture we made in the p=3 case.
Stichwort
Infinite Borwein productSign patternAsymptoticsPositivityCircle methodVanishing of coefficients
Objekt-Typ
Sprache
Englisch [eng]
Persistent identifier
Erschienen in
Titel
The Ramanujan Journal
ISSN
1382-4090
Erscheinungsdatum
2021
Publication
Springer Science and Business Media LLC
Erscheinungsdatum
2021
Zugänglichkeit
Rechte
Rechteangabe
© The Author(s) 2021
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