Generalized Heine’s identity for complex Fourier series of binomials

Abstract

In his treatise, Heine ( Heine 1881 In Theorie und Anwendungen ) gave an identity for the Fourier series of the function , with , and z >1, in terms of associated Legendre functions of the second kind . In this paper, we generalize Heine’s identity for the function , with , and , in terms of . We also compute closed-form expressions for some .

Legendre functions of fractional degree: transformations and evaluations

“…This is the little known Fourier expansion of (1 + x cos φ) ν (cf. [5,14]), and can be written as a Fourier cosine series. It follows from this representation that if ν = − 1 4 or ν = − 1 6 , or more generally if ν differs from an integer by ±1/r with r = 3, 4, 6 (as well as the classical case r = 2), the Fourier coefficients of (1 + x cos φ) ν can be expressed in terms of complete elliptic integrals.…”

“…VII, §7.3]. Equation (1.1b) is a 'generalized Heine identity' which has attracted attention [5], and can be viewed as an analytic continuation of (1.1a). When ν = − 1 2 , it leads to an alternative to the usual multipole expansion of the 1/|x − x | potential [6,7].…”

Fourier expansions for a logarithmic fundamental solution of the polyharmonic equation

“…Solutions to inhomogeneous polyharmonic equations are useful in many physical applications including those areas related to Poisson's equation such as Newtonian gravity, electrostatics, magnetostatics, quantum direct and exchange interactions (cf. §1 in Cohl & Dominici (2010) [4]), etc. Furthermore, applications of higher-powers of the Laplacian include such varied areas as minimal surfaces [13], Continuum Mechanics [9], Mesh deformation [7], Elasticity [10], Stokes Flow [8], Geometric Design [20], Cubature formulae [17], mean value theorems (cf.…”

“…The first approach is algebraic and involves the generation of a certain set of naturally arising two-index polynomials which we refer to as logarithmic polynomials. The second approach starts with the main result from Cohl & Dominici (2010) [4] and determines the Fourier series expansion for a logarithmic fundamental solution of the polyharmonic equation through parameter differentiation. Series expansions for fundamental solutions of linear partial differential equations such as the polyharmonic equation are extremely useful in determining Dirichlet boundary values for solutions on interior domains (see for example Cohl & Tohline (1999) [5]).…”

“…∞) → C appearing in[4] to the associated Legendre function of the first kind P µ ν : (1, ∞) → R. The associated Legendre function of the first kind can be defined using the Gauss hypergeometric function, namely(Magnus, Oberhettinger & Soni (1966) [11], p. 153)…”