{"title":"Two general series identities involving modified Bessel functions and a class of arithmetical functions","authors":"B. Berndt, A. Dixit, Rajat Gupta, A. Zaharescu","doi":"10.4153/S0008414X22000530","DOIUrl":null,"url":null,"abstract":"Abstract We consider two sequences \n$a(n)$\n and \n$b(n)$\n , \n$1\\leq n<\\infty $\n , generated by Dirichlet series \n$$ \\begin{align*}\\sum_{n=1}^{\\infty}\\frac{a(n)}{\\lambda_n^{s}}\\qquad\\text{and}\\qquad \\sum_{n=1}^{\\infty}\\frac{b(n)}{\\mu_n^{s}},\\end{align*} $$\n satisfying a familiar functional equation involving the gamma function \n$\\Gamma (s)$\n . Two general identities are established. The first involves the modified Bessel function \n$K_{\\mu }(z)$\n , and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are \n$K_{\\mu }(z)$\n , the Bessel functions of imaginary argument \n$I_{\\mu }(z)$\n , and ordinary hypergeometric functions \n${_2F_1}(a,b;c;z)$\n . Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function \n$\\tau (n)$\n , the number of representations of n as a sum of k squares \n$r_k(n)$\n , and primitive Dirichlet characters \n$\\chi (n)$\n .","PeriodicalId":55284,"journal":{"name":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","volume":"101 1","pages":"1800 - 1830"},"PeriodicalIF":0.6000,"publicationDate":"2022-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Journal of Mathematics-Journal Canadien De Mathematiques","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4153/S0008414X22000530","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 4
Abstract
Abstract We consider two sequences
$a(n)$
and
$b(n)$
,
$1\leq n<\infty $
, generated by Dirichlet series
$$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$
satisfying a familiar functional equation involving the gamma function
$\Gamma (s)$
. Two general identities are established. The first involves the modified Bessel function
$K_{\mu }(z)$
, and can be thought of as a ‘modular’ or ‘theta’ relation wherein modified Bessel functions, instead of exponential functions, appear. Appearing in the second identity are
$K_{\mu }(z)$
, the Bessel functions of imaginary argument
$I_{\mu }(z)$
, and ordinary hypergeometric functions
${_2F_1}(a,b;c;z)$
. Although certain special cases appear in the literature, the general identities are new. The arithmetical functions appearing in the identities include Ramanujan’s arithmetical function
$\tau (n)$
, the number of representations of n as a sum of k squares
$r_k(n)$
, and primitive Dirichlet characters
$\chi (n)$
.
期刊介绍:
The Canadian Journal of Mathematics (CJM) publishes original, high-quality research papers in all branches of mathematics. The Journal is a flagship publication of the Canadian Mathematical Society and has been published continuously since 1949. New research papers are published continuously online and collated into print issues six times each year.
To be submitted to the Journal, papers should be at least 18 pages long and may be written in English or in French. Shorter papers should be submitted to the Canadian Mathematical Bulletin.
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