  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].
