{"title":"On the infinite Borwein product raised to a positive real power.","authors":"Michael J Schlosser, Nian Hong Zhou","doi":"10.1007/s11139-021-00519-3","DOIUrl":null,"url":null,"abstract":"<p><p>In this paper, we study properties of the coefficients appearing in the <i>q</i>-series expansion of <math><mrow><msub><mo>∏</mo><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><msup><mrow><mo>[</mo><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>q</mi><mi>n</mi></msup><mo>)</mo></mrow><mo>/</mo><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>q</mi><mrow><mi>pn</mi></mrow></msup><mo>)</mo></mrow><mo>]</mo></mrow><mi>δ</mi></msup></mrow></math>, the infinite Borwein product for an arbitrary prime <i>p</i>, raised to an arbitrary positive real power <math><mi>δ</mi></math>. We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients. For <math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math> 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 <math><mi>δ</mi></math> 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 <math><mrow><mi>p</mi><mo>=</mo><mn>3</mn></mrow></math> case.</p>","PeriodicalId":54511,"journal":{"name":"Ramanujan Journal","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10185621/pdf/","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ramanujan Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11139-021-00519-3","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2021/11/2 0:00:00","PubModel":"Epub","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
In this paper, we study properties of the coefficients appearing in the q-series expansion of , the infinite Borwein product for an arbitrary prime p, raised to an arbitrary positive real power . We use the Hardy-Ramanujan-Rademacher circle method to give an asymptotic formula for the coefficients. For 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 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 case.
期刊介绍:
The Ramanujan Journal publishes original papers of the highest quality in all areas of mathematics influenced by Srinivasa Ramanujan. His remarkable discoveries have made a great impact on several branches of mathematics, revealing deep and fundamental connections.
The following prioritized listing of topics of interest to the journal is not intended to be exclusive but to demonstrate the editorial policy of attracting papers which represent a broad range of interest:
Hyper-geometric and basic hyper-geometric series (q-series) * Partitions, compositions and combinatory analysis * Circle method and asymptotic formulae * Mock theta functions * Elliptic and theta functions * Modular forms and automorphic functions * Special functions and definite integrals * Continued fractions * Diophantine analysis including irrationality and transcendence * Number theory * Fourier analysis with applications to number theory * Connections between Lie algebras and q-series.