Generalized Laplace coefficients and Newcomb derivatives

Abstract

It is shown how the generalized Laplace coefficients can be employed to deduce explicit formulas for ordinary and Newcomb derivatives of the Laplace coefficients.

Generalized Heine’s identity for complex Fourier series of binomials

“…Algebraic functions of the form (z − cos j) −m arise naturally in classical physics through the theory of fundamental solutions of Laplace's equation, and they represent powers of distance between two points in a Euclidean space. Their expansions in Fourier series have a rich history, appearing in the theory of arbitrarily shaped charge distributions in electrostatics (Barlow 2003;Popsueva et al 2007;Pustovitov 2008a,b;Verdú et al 2008), magnetostatics (Selvaggi et al 2008b;Beleggia et al 2009), quantum direct and exchange Coulomb interactions (Cohl et al 2001;Enriquez & Blum 2005;Gattobigio et al 2005;Poddar & Deb 2007;Bagheri & Ebrahimi 2008), Newtonian gravity (Fromang 2005;Huré 2005;Huré & Pierens 2005;Chan et al 2006;Ou 2006;Saha & Jog 2006;Boley & Durisen 2008;Mellon & Li 2008;Selvaggi et al 2008a;Even & Tohline 2009;Schachar et al 2009), the Laplace coefficients of the planetary disturbing function (D'Eliseo 1989(D'Eliseo , 2007, and potential fluid flow around actuator discs (Hough & Ordway 1965;Breslin & Andersen 1994), just to name a few physical applications. A precise Fourier analysis is extremely useful to fully describe the general non-axisymmetric nature of these problems.…”