The generating functions of Lame equation in Weierstrass's form

Abstract

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 B_n term terminated [29], I consider generating function of it including all higher terms of A_n'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].

20 pages, final version. arXiv admin note: substantial text overlap with arXiv:1303.0878, arXiv:1310.7811, arXiv:1303.0873, arXiv:1303.0819, arXiv:1303.0813, arXiv:1303.0876, arXiv:1303.0820, arXiv:1302.7309

References in corpus (7)

• Generalization of the three-term recurrence formula and its applications
• Asymptotic behavior of Heun function and its integral formalism
• Analytic solution for grand confluent hypergeometric function
• The integral formalism and the generating function of grand confluent hypergeometric function
• Lame equation in the algebraic form
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Power series and integral forms of Lame equation in Weierstrass's form

Cited by in corpus (8)

• Asymptotic behavior of Heun function and its integral formalism
• Generalization of the three-term recurrence formula and its applications
• Analytic solution for grand confluent hypergeometric function
• The integral formalism and the generating function of grand confluent hypergeometric function
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Lame equation in the algebraic form
• Power series and integral forms of Lame equation in Weierstrass's form
• Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic Pöschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well