Generalization of the three-term recurrence formula and its applications

Abstract

The history of linear differential equations is over 350 years. By using Frobenius method and putting the power series expansion into linear differential equations, the recursive relation of coefficients starts to appear. There can be between two and infinity number of coefficients in the recurrence relation in the power series expansion. During this period mathematicians developed analytic solutions of only two term recursion relation in closed forms. Currently the analytic solution of three term recurrence relation is unknown. In this paper I will generalize the three term recurrence relation in the linear differential equation. This paper is 2nd out of 10 in series "Special functions and three term recurrence formula (3TRF)". The next paper in series deals with the power series expansion in closed forms of Heun function by Choun [arXiv:1303.0830]. The rest of the papers in the series show how to solve mathematical equations having three term recursion relations and go on producing the exact solutions of some of the well known special functions including: Mathieu, Heun, Biconfluent Heun and Lame equations. See section IX for all the papers and short descriptions in the series.

21 pages, final version. arXiv admin note: substantial text overlap with arXiv:1310.7811, arXiv:1303.0873

References in corpus (9)

• Heun Functions and Some of Their Applications in Physics
• Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces
• Asymptotic behavior of Heun function and its integral formalism
• The integral formalism and the generating function of grand confluent hypergeometric function
• Analytic solution for grand confluent hypergeometric function
• The generating functions of Lame equation in Weierstrass's form
• 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 (10)

• Heun Functions and Some of Their Applications in Physics
• Asymptotic behavior of Heun function and its integral formalism
• Analytic solution for grand confluent hypergeometric function
• Lame equation in the algebraic form
• The integral formalism and the generating function of grand confluent hypergeometric function
• The generating functions of Lame equation in Weierstrass's 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
• Special functions and reversible three-term recurrence formula (R3TRF)
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)