Lame equation in the algebraic form

Abstract

Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame functions are applicable to diverse areas such as boundary value problems in ellipsoidal geometry, chaotic Hamiltonian systems, the theory of Bose-Einstein condensates, etc. In this paper I will apply three term recurrence formula [arXiv:1303.0806] to the power series expansion in closed forms of Lame function in the algebraic form(infinite series and polynomial) and its integral forms including all higher terms of A_n's. I will show how to transform power series expansion of Lame function to an integral formalism mathematically for cases of infinite series and polynomial. One interesting observation resulting from the calculations is the fact that a Gauss Hypergeometric function recurs in each of sub-integral forms: the first sub-integral form contains zero term of A_n's, the second one contains one term of A_n's, the third one contains two terms of A_n's, etc. Section 6 contains additional examples of application in Lame function. This paper is 6th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 7 for all the papers in the series. Previous paper in series deals with the power series expansion of Mathieu function and its integral formalism [arXiv:1303.0820]. The next paper in the series describes the power series and integral forms of Lame equation in the Weierstrass's form and its asymptotic behaviors [arXiv:1303.0878].