Lame equation in the algebraic form

2013/03/04 by Yoon-Seok Choun, Yoon Seok Choun, Choun, Yoon Seok · 9 citations
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Quantum chaos and dynamical systems #Scientific Research and Discoveries #math-ph #math.CA #math.MP

paper · pdf · doi:10.48550/arxiv.1303.0873

26 pages, Typos corrected in equations in sec.3. arXiv admin note: substantial text overlap with arXiv:1303.0820, arXiv:1303.0819, arXiv:1303.0806

openalex publication_date 2013/03/04 · arxiv created 2015/06/27 · arxiv updated 2015/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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].

Citations

• Analytic solution for grand confluent hypergeometric function
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• The generating functions of Lame equation in Weierstrass's form
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)
• Generalization of the three-term recurrence formula and its applications
• The power series expansion of Mathieu function and its integral formalism
• Power series and integral forms of Lame equation in Weierstrass's form
• The integral formalism and the generating function of grand confluent hypergeometric function

Cited by

• Generalization of the three-term recurrence formula and its applications
• The generating functions of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)
• Analytic solution for grand confluent hypergeometric function
• The integral formalism and the generating function of grand confluent hypergeometric function
• The power series expansion of Mathieu function and its integral formalism
• The analytic solution for the power series expansion of Heun function
• Asymptotic behavior of Heun function and its integral formalism
• Power series and integral forms of Lame equation in Weierstrass's form

Related

• Power series and integral forms of Lame equation in Weierstrass's form
• The generating functions of Lame equation in Weierstrass's form
• Special functions and reversible three-term recurrence formula (R3TRF)
• Analytic solution for grand confluent hypergeometric function
• Generalization of the three-term recurrence formula and its applications