The generating functions of Lame equation in Weierstrass's form

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Lame equation arises from deriving Laplace equation in ellipsoidal coordinates; in other words, it's called ellipsoidal harmonic equation. Lame function is applicable to diverse areas such as boundary value problems in ellipsoidal geometry, chaotic Hamiltonian systems, the theory of Bose-Einstein condensates, etc. By applying generating function into modern physics (quantum mechanics, thermodynamics, black hole, supersymmetry, special functions, etc), we are able to obtain the recursion relation, a normalization constant for the wave function and expectation values of any physical quantities. For the case of hydrogen-like atoms, generating function of associated Laguerre polynomial has been used in order to derive expectation values of position and momentum. By applying integral forms of Lame polynomial in the Weierstrass's form in which makes Bₙ term terminated [29], I consider generating function of it including all higher terms of Aₙ's. This paper is 8th out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 4 for all the papers in the series. Previous paper in series deals with the power series expansion and the integral formalism of Lame equation in the Weierstrass's form and its asymptotic behavior [29]. The next paper in the series describes analytic solution for grand confluent hypergeometric function [31].