Two general series identities involving modified Bessel functions and a class of arithmetical functions

IF 0.6 3区 数学 Q3 MATHEMATICS
B. Berndt, A. Dixit, Rajat Gupta, A. Zaharescu
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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 $\chi (n)$ .
涉及修正贝塞尔函数和一类算术函数的两个一般级数恒等式
摘要考虑由Dirichlet级数$$ \begin{align*}\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},\end{align*} $$生成的两个序列$a(n)$和$b(n)$, $1\leq n<\infty $,满足一个熟悉的包含gamma函数$\Gamma (s)$的泛函方程。建立了两个一般的恒等式。第一个涉及修改的贝塞尔函数$K_{\mu }(z)$,可以被认为是一个“模”或“θ”关系,其中出现了修改的贝塞尔函数,而不是指数函数。在第二个恒等式中出现的有$K_{\mu }(z)$、虚参的贝塞尔函数$I_{\mu }(z)$和一般的超几何函数${_2F_1}(a,b;c;z)$。虽然在文献中出现了一些特殊的情况,但一般的身份是新的。在恒等式中出现的算术函数包括拉马努金的算术函数$\tau (n)$,表示n为k平方和的个数$r_k(n)$,以及原始狄利克雷字符$\chi (n)$。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
58
审稿时长
4.5 months
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. Le Journal canadien de mathématiques (JCM) publie des articles de recherche innovants de grande qualité dans toutes les branches des mathématiques. Publication phare de la Société mathématique du Canada, il est publié en continu depuis 1949. En ligne, la revue propose constamment de nouveaux articles de recherche, puis les réunit dans des numéros imprimés six fois par année. Les textes présentés au JCM doivent compter au moins 18 pages et être rédigés en anglais ou en français. C’est le Bulletin canadien de mathématiques qui reçoit les articles plus courts.
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