代数形式的拉梅方程
Lame equation in the algebraic form
摘要：
拉梅方程是在椭球坐标系下推导拉普拉斯方程而得到的，即椭球调和方程。Lame函数可应用于不同的领域，如椭球几何中的边值问题，混沌Hamilton系统，玻色-爱因斯坦凝聚体理论等。本文将应用三项递推公式[arXiv：1303.0806]对代数形式（无穷级数和多项式）的Lame函数及其包含所有高阶项的积分形式的封闭形式的幂级数展开。我将展示如何将拉梅函数的幂级数展开转换为无穷级数和多项式情况下的数学积分形式。从计算中得到的一个有趣的观察结果是，高斯超几何函数在每一个子积分形式中递归：第一个子积分形式包含A_n的零项，第二个子积分形式包含A_n的一项，第三个子积分形式包含A_n的两项，等等。本文是“特殊函数与三项递推公式（3TRF）”系列文章10篇之6。见第7节的所有文件在该系列。上一篇系列论文涉及Mathieu函数的幂级数展开及其积分形式[arXiv：1303.0820]。该系列的下一篇论文描述了Weierstrass形式的Lame方程的幂级数和积分形式及其渐近行为[arXiv：1303.0878]。
英文摘要：
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].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
