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Two general series identities involving modified Bessel functions and a class of arithmetical functions
- Part of
-
- Journal:
- Canadian Journal of Mathematics / Volume 75 / Issue 6 / December 2023
- Published online by Cambridge University Press:
- 10 October 2022, pp. 1800-1830
- Print publication:
- December 2023
-
- Article
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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 
$\chi (n)$.
