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2017 | OriginalPaper | Buchkapitel

4. Schlömilch Series

verfasst von : Árpád Baricz, Dragana Jankov Maširević, Tibor K. Pogány

Erschienen in: Series of Bessel and Kummer-Type Functions

Verlag: Springer International Publishing

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Abstract

This chapter is devoted to the study of integral representations of Schlömilch series built by Bessel functions of the first kind and modified Bessel functions of the second kind. Closed expressions for some special Schlömilch series together with their connection to Mathieu series are also investigated. The chapter ends with an integral representation formula for number theoretical summation by Popov, which also covers the theta-transform identity coming from functional equation for the Epstein Zeta function.

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Vorheriges Kapitel Kapteyn Series

Nächstes Kapitel Miscellanea

Moreover, Berndt et al. extended (4.7) to all 2q > k − 3, compare [35, Theorem 2.1].

Actually, A _p,q(γ) is the Laplace transform of $x \mapsto x^{-1} \mathrm {e}^{-\frac px} J_\nu (\gamma x)$ at the argument q.

The usually used integral expression (2.4) for the Bessel function in the summands of (4.41) results in

$$\displaystyle \begin{aligned} \mathfrak S_{k, q}(x) = \frac{2 (\pi x)^{\frac{k}{2} + q}}{ \sqrt{\pi}\,\varGamma\left( \frac{k+1}2 + q\right)} \, \int_0^1 (1-t^2)^{\frac{k-1}2 + q}\, \sum_{n \geq 1} r_k(n)\,\cos\left(2\pi t \sqrt{nx}\right)\, \mathrm{d}t.\end{aligned}$$

On the other hand, also by Walfisz was found that [326, p. 40]

$$\displaystyle \begin{aligned} \sum_{j = 1}^n r_k(n) = cn^{\frac{k}{2}} + \mathscr O\big(n^{\frac{k-1}2}\big)\, ,\end{aligned}$$

being c an absolute constant. All together imply that the inner sum diverges in a neighborhood of t = 0, therefore the integral diverges too.

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Titel: Schlömilch Series
verfasst von: Árpád Baricz
Dragana Jankov Maširević
Tibor K. Pogány
Verlag: Springer International Publishing
Buch: Series of Bessel and Kummer-Type Functions
Print ISBN: 978-3-319-74349-3

Electronic ISBN: 978-3-319-74350-9

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-74350-9_4

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