{"title":"On some coefficients of the Artin–Hasse series modulo a prime","authors":"Marina Avitabile, Sandro Mattarei","doi":"10.1016/j.indag.2024.01.003","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>p</mi></math></span> be an odd prime, and let <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>[</mo><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>]</mo></mrow></mrow></math></span> be the reduction modulo <span><math><mi>p</mi></math></span> of the Artin–Hasse exponential series. We obtain a polynomial expression for <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mi>p</mi></mrow></msub></math></span> in terms of those <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>r</mi><mi>p</mi></mrow></msub></math></span> with <span><math><mrow><mi>r</mi><mo><</mo><mi>k</mi></mrow></math></span>, for even <span><math><mrow><mi>k</mi><mo><</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>1</mn></mrow></math></span>. A conjectural analogue covering the case of odd <span><math><mrow><mi>k</mi><mo><</mo><mi>p</mi></mrow></math></span> can be stated in various polynomial forms, essentially in terms of the polynomial <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>n</mi><mo>)</mo></mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi><mo>−</mo><mi>n</mi></mrow></msup></mrow></math></span>, where <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denotes the <span><math><mi>n</mi></math></span>th Bernoulli number.</p><p>We prove that <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> satisfies the functional equation <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>−</mo><mi>γ</mi><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow><mi>£</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>−</mo><mn>1</mn></mrow></math></span> in <span><math><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow></mrow></math></span>, where <span><math><mrow><msub><mrow><mi>£</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> are the truncated logarithm and the Wilson quotient. This is an analogue modulo <span><math><mi>p</mi></math></span> of a functional equation, in <span><math><mrow><mi>Q</mi><mrow><mo>[</mo><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mo>]</mo></mrow></mrow></math></span>, established by Zagier for the power series <span><math><mrow><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>∞</mi></mrow></msubsup><mrow><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>/</mo><mi>n</mi><mo>)</mo></mrow><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span>. The proof of our functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.5000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Indagationes Mathematicae-New Series","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S001935772400003X","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be an odd prime, and let be the reduction modulo of the Artin–Hasse exponential series. We obtain a polynomial expression for in terms of those with , for even . A conjectural analogue covering the case of odd can be stated in various polynomial forms, essentially in terms of the polynomial , where denotes the th Bernoulli number.
We prove that satisfies the functional equation in , where and are the truncated logarithm and the Wilson quotient. This is an analogue modulo of a functional equation, in , established by Zagier for the power series . The proof of our functional equation establishes a connection with a result of Nielsen of 1915, of which we provide a fresh proof. Our polynomial framing allows us to derive congruences for certain numerical sums involving divided Bernoulli numbers.
期刊介绍:
Indagationes Mathematicae is a peer-reviewed international journal for the Mathematical Sciences of the Royal Dutch Mathematical Society. The journal aims at the publication of original mathematical research papers of high quality and of interest to a large segment of the mathematics community. The journal also welcomes the submission of review papers of high quality.