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Applications of the Lipschitz Summation Formula and a Generalization of Raabe’s Cosine Transform

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Constructive Approximation Aims and scope

Abstract

General summation formulas have been proved to be very useful in analysis, number theory and other branches of mathematics. The Lipschitz summation formula is one of them. In this paper, we give its application by providing a new transformation formula which generalizes that of Ramanujan. Ramanujan’s result, in turn, is a generalization of the modular transformation of Eisenstein series \(E_k(z)\) on SL\(_2({\mathbb {Z}})\), where \(z\rightarrow -1/z, z\in {\mathbb {H}}\). The proof of our result involves delicate analysis containing Cauchy Principal Value integrals. A simpler proof of a recent result of ours with Kesarwani giving a non-modular transformation for \(\sum _{n=1}^{\infty }\sigma _{2m}(n)e^{-ny}\) is also derived using the Lipschitz summation formula. In the pursuit of obtaining this transformation, we naturally encounter a new generalization of Raabe’s cosine transform whose several properties are also demonstrated. As an application of our results, we get a generalization of Wright’s asymptotic estimate for the generating function of the number of plane partitions of a positive integer n.

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Notes

  1. Ramanujan’s formula is actually valid for any complex \(\alpha , \beta \) such that \(Re (\alpha )>0, Re (\beta )>0\) and \(\alpha \beta =\pi ^2\).

  2. The case \(a=0\) of (3.16) reduces to a result of Kuylenstierna [32, Equation (7)].

  3. Note that one has to justify the interchange of the order of the integrals in third step of [39, p. 83] to use Parseval’s formula. The conditions under which it can be done are given after [39, p. 83, Equation (3.1.11)]. But one of our integrals is a principal value integral, therefore, we need to justify this interchange of the order of the integration.

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Acknowledgements

The authors sincerely thank referees for giving several important suggestions which enhanced the exposition of the paper. Further, they thank George E. Andrews for a helpful discussion on plane partitions. They would also like to thank Donghun Yu from POSTECH library for arranging references [26, 27, 50] for them. The first author’s research was partially supported by the Swarnajayanti Fellowship Grant SB/SJF/2021-22/08 of SERB (Govt. of India) and the CRG grant CRG/2020/002367 of SERB. The second author’s research was supported by the grant IBS-R003-D1 of the IBS-CGP, POSTECH, South Korea, and the Fulbright-Nehru Postdoctoral Fellowship Grant 2846/FNPDR/2022. Both the authors sincerely thank the respective funding agencies for their support. Part of this work was done when the second author was visiting IIT Gandhinagar in May 2022. He sincerely thanks IBS-CGP for the financial support and IIT Gandhinagar for its hospitality.

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Author notes

  1. Rahul Kumar

    Present address: Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, Gujarat, 382355, India

    Atul Dixit

  2. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Republic of Korea

    Rahul Kumar

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Dixit, A., Kumar, R. Applications of the Lipschitz Summation Formula and a Generalization of Raabe’s Cosine Transform. Constr Approx (2023). https://doi.org/10.1007/s00365-023-09668-8

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