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Second Type Neumann Series Related to Nicholson’s and to Dixon–Ferrar Formula

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Transmutation Operators and Applications

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Abstract

The second type Neumann series are considered which building blocks are Nicholson’s and the Dixon–Ferrar formulae for \(J_\nu ^2(x)+ Y_\nu ^2(x)\). Related closed form double definite integral expressions are established by using the associated Dirichlet’s series Cahen’s Laplace integral for the Nicholson’s case. However, using Dixon–Ferrar formula a double definite integral expression is again obtained. Certain Open Problems are posed in the last section of the chapter.

Dedicated to Gradimir V. Milovanović to his 70th birthday anniversary

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Notes

  1. 1.

    Instead of the determinant form, we will use the most popular cross–product expression J ν(x)Y μ(x) − J μ(x)Y ν(x) throughout.

  2. 2.

    The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series, while Maclaurin used it to calculate integrals.

  3. 3.

    Here, and in what follows 1A denotes the indicator function of the set A.

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Acknowledgements

T.K. Pogány acknowledges the support given by the NAWA project PROM PPI/PRO/2018/1/00008 and thanks to the Department of Mathematical Physics, The Henryk Niewodniczański Institute of Nuclear Physics of Polish Academy of Sciences, Kraków, Poland for the warm hospitality and the excellent working atmosphere during his stay there during February 2019.

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Cvijović, D., Pogány, T.K. (2020). Second Type Neumann Series Related to Nicholson’s and to Dixon–Ferrar Formula. In: Kravchenko, V., Sitnik, S. (eds) Transmutation Operators and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-35914-0_4

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