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Schlömilch Series

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Series of Bessel and Kummer-Type Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2207))

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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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Notes

  1. 1.

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

  2. 2.

    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.

  3. 3.

    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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Authors and Affiliations

  1. John von Neumann Faculty of Informatics, Institute of Applied Mathematics, Óbuda University, Budapest, Hungary

    Árpád Baricz & Tibor K. Pogány

  2. Department of Economics, Babeş–Bolyai University, Cluj–Napoca, Romania

    Árpád Baricz

  3. Department of Mathematics, Josip Juraj Strossmayer University of Osijek, Osijek, Croatia

    Dragana Jankov Maširević

  4. Faculty of Maritime Studies, University of Rijeka, Rijeka, Croatia

    Tibor K. Pogány

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Baricz, Á., Jankov Maširević, D., Pogány, T.K. (2017). Schlömilch Series. In: Series of Bessel and Kummer-Type Functions. Lecture Notes in Mathematics, vol 2207. Springer, Cham. https://doi.org/10.1007/978-3-319-74350-9_4

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