Second Type Neumann Series of Generalized Nicholson Function
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The second type Neumann series are considered whose building blocks are generalized Nicholson’s functions \(B_\nu ^p(x) :=J_\nu ^p(x)+ Y_\nu ^p(x)\), being \(J_\nu , Y_\nu \) Bessel functions of the first and second kind of order \(\nu \), \(p \ge 2\) integer. Closed form definite integral expressions are obtained for such series with the aid of the associated Dirichlet series’ Cahen’s Laplace integral form.
KeywordsBessel functions of the first and second kind modified Bessel function of the second kind \(K_\nu \) Nicholson function Neumann series of Bessel functions Cahen’s Laplace integral formula
Mathematics Subject ClassificationPrimary 33C10 33E20 40C10 Secondary 33E30 40H05
The authors are indebted to both unknown referees for several constructive comments which mainly improve the exposition’s relevance and completeness and finally encompass the article. 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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