Research Article Connection Formulae between Ellipsoidal and Spherical Harmonics with Applications to Fluid Dynamics and Electromagnetic Scattering Michael Doschoris 1,2 and Panayiotis Vafeas 1 1 Department of Chemical Engineering, University of Patras, 26504 Patras, Greece 2 Institute of Chemical Engineering Sciences, Stadiou Street, P.O. Box 1414, Platani, 26504 Patras, Greece Correspondence should be addressed to Michael Doschoris; mdoscho@chemeng.upatras.gr Received 26 July 2014; Accepted 30 October 2014 Academic Editor: George Dassios Copyright © 2015 M. Doschoris and P. Vafeas. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. he environment of the ellipsoidal system, signiicantly more complex than the spherical one, provides the necessary settings for tackling boundary value problems in anisotropic space. However, the theory of Lam´ e functions and ellipsoidal harmonics ailiated with the ellipsoidal system is rather complicated. A turning point would reside in the existence of expressions interlacing these two diferent systems. Still, there is no simple way, if at all, to bridge the gap. he present paper addresses this issue. We provide explicit formulas of speciic ellipsoidal harmonics expressed in terms of their counterparts in the classical spherical system. hese expressions are then put into practice in the framework of physical applications. 1. Introduction he ellipsoidal coordinate system, by its very nature, is demanding concealing numerous diiculties. he main rea- son can be associated with the acquisition of solutions for miscellaneous operators. Even in the case of the Laplacian, deriving the corresponding eigensolutions is a nontrivial task. he French engineer and mathematician Gabriel Lam´ e in the mid nineteenth century, following an ingenious argument, separated variables for the Laplace operator arriving at the functions, which carry nowadays his name. Taking the product of Lam´ e functions leads to the ellipsoidal harmonics. But the complications regarding the particular system do not end here. In contrast to the theory of spherical harmonics, only ellipsoidal harmonics of low order have been computed in closed form [1, 2]. Why? First of all, a recursive technique in order to generate Lam´ e functions does not exist. Although we know that Lam´ e functions are connected by three-term recurrence relations [3], to the authors knowledge no procedure calculating the corresponding coeicients has been proposed so far. his particular impediment forces us to undergo an involved algorithm from which the Lam´ e functions are determined. his essentially two-step opera- tion requires the computation of the roots of polynomial functions allowing nontrivial solutions for the initial linear homogeneous systems. We note that the previously indicated algorithm can be applied analytically only for Lam´ e functions up to the seventh degree [2]. Higher-order terms demand computational implementations [4] introducing numerical instability, which in the sequence is transferred to the calcu- lation of the corresponding Lam´ e function. he aforementioned hurdles could in theory be avoided on the assumption that the functions of Lam´ e and corre- sponding ellipsoidal harmonics would be able to be expressed in terms of Legendre functions and spherical harmonics, respectively. Although, in principle, the possibility exists, no general formulae connecting these functions are available. he absence of such relations is justiied bearing in mind that ellipsoidal harmonics are not reducible in a straightforward and unique way to the corresponding spherical harmonics. Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2015, Article ID 572458, 12 pages http://dx.doi.org/10.1155/2015/572458