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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.)
Software:
DLMF
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