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On the infinite Borwein product raised to a positive real power
- Source :
- The Ramanujan Journal. 61:515-543
- Publication Year :
- 2021
- Publisher :
- Springer Science and Business Media LLC, 2021.
-
Abstract
- In this paper, we study properties of the coefficients appearing in the $q$-series expansion of $\prod_{n\ge 1}[(1-q^n)/(1-q^{pn})]^\delta$, the infinite Borwein product for an arbitrary prime $p$, raised to an arbitrary positive real power $\delta$. 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 $\delta$ 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.<br />Comment: 24 pages; an appendix with several new conjectures added; to appear in the Ramanujan Journal; dedicated to the memory of Richard Allen Askey
- Subjects :
- 11P55 (Primary) 11F03, 11F30, 26D20 (Secondary)
Algebra and Number Theory
Conjecture
Mathematics - Number Theory
Vanishing of coefficients
010102 general mathematics
Infinite product
Positivity
Divisibility rule
Infinite Borwein product
01 natural sciences
Circle method
010101 applied mathematics
Combinatorics
Number theory
Product (mathematics)
FOS: Mathematics
Asymptotic formula
Number Theory (math.NT)
0101 mathematics
Positive real numbers
Sign pattern
Asymptotics
Sign (mathematics)
Mathematics
Subjects
Details
- ISSN :
- 15729303 and 13824090
- Volume :
- 61
- Database :
- OpenAIRE
- Journal :
- The Ramanujan Journal
- Accession number :
- edsair.doi.dedup.....0a1d70c5441182d28641523f6a945462