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Asymptotics of some generalized Mathieu series. (English) Zbl 07276082
Let $$\mu\ge0$$ and $$r>0$$, and fix two series $$\mathbf{a}=(a_n)_{n\ge0}$$, $$\mathbf{b}=(b_n)_{n\ge0}$$. The generalized Mathieu series is defined as $S_{\mathrm{a}, \mathrm{b}, \mu}(r):=\sum_{n=0}^{\infty} \frac{a_{n}}{\left(b_{n}+r^{2}\right)^{\mu+1}}.$ Moreover, for $$\alpha, \beta, r>0, \mu \geq 0,$$ with $$\alpha-\beta(\mu+1)<-1$$ and $$\gamma, \delta \in \mathbb{R}$$ the series $S_{\alpha, \beta, \gamma, \delta, \mu}(r):=\sum_{n=2}^{\infty} \frac{n^{\alpha}(\log n)^{\gamma}}{\left(n^{\beta}(\log n)^{\delta}+r^{2}\right)^{\mu+1}}$ is defined. Another special case of the generalized Mathieu series is $S_{\alpha, \beta, \mu}^{!}(r):=\sum_{n=0}^{\infty} \frac{(n !)^{\alpha}}{\left((n !)^{\beta}+r^{2}\right)^{\mu+1}},$ for appropriate parameter choice.
The authors then study the asymptotic behavior of these series. Among others, it turns out that $S_{\alpha, \beta, \gamma, \delta, \mu}(r) \sim C_{\alpha, \beta, \gamma, \delta, \mu} r^{2(\alpha+1) / \beta-2(\mu+1)}(\log r)^{-\delta(\alpha+1) / \beta+\gamma}, \quad r \uparrow \infty,$ again, for appropriate parameter choice. Similar asymptotics hold for $$S_{\alpha, \beta, \mu}^{!}(r)$$, too. Here $$C_{\alpha, \beta, \gamma, \delta, \mu}$$ is an expression involving the Gamma function.
##### MSC:
 33E20 Other functions defined by series and integrals 11M41 Other Dirichlet series and zeta functions 44A15 Special integral transforms (Legendre, Hilbert, etc.)
DLMF
Full Text:
##### References:
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