Special functions and reversible three-term recurrence formula (R3TRF)

2013/09/27 by Yoon-Seok Choun, Yoon Seok Choun, Choun, Yoon Seok
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Matrix Theory and Algorithms #math.CA

paper · pdf · doi:10.48550/arxiv.1310.7811

486 pages, final version. arXiv admin note: substantial text overlap with arXiv:1405.3610, arXiv:1303.0876, arXiv:1303.0873, arXiv:1303.0878, arXiv:1303.0820, arXiv:1303.0879, arXiv:1303.0806, arXiv:1303.0819

openalex publication_date 2013/09/27 · arxiv created 2014/11/04 · arxiv updated 2014/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In the previous series "Special functions and three term recurrence formula (3TRF)", I generalize the three term recurrence relation in the linear differential equation for the infinite series and polynomial which makes B_n term terminated including all higher terms of A_n's. In this series I will show how to obtain the formula for the polynomial which makes A_n term terminated including all higher terms of B_n's and infinite series of its power series expansion. In the future series I will show you for the polynomial which makes A_n and B_n terms terminated at same time; the power series, integral formalism and generating function such as Heun, Mathieu, Lame and GCH equations will be constructed analytically. In chapter 1, I will generalize the three term recurrence relation in linear differential equation in a backward for the infinite series and polynomial which makes A_n term terminated including all higher terms of B_n's. In chapters 2-9, I will apply reversible three term recurrence formula to (1) the power series expansion in closed forms, (2) its integral representation and (3) generating functions of Heun, Confluent Heun, GCH, Lame and Mathieu equations that consist of three term recursion relation for the infinite series and polynomial which makes A_n term terminated.

Citations

• Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations
• Analytic solution for grand confluent hypergeometric function
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)
• Asymptotic behavior of Heun function and its integral formalism
• Lame equation in the algebraic form
• Power series and integral forms of Lame equation in Weierstrass's form
• The generating functions of Lame equation in Weierstrass's form
• Generalization of the three-term recurrence formula and its applications
• 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
• 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

Related

• Generalization of the three-term recurrence formula and its applications
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)
• Analytic solution for grand confluent hypergeometric function
• Power series and integral forms of Lame equation in Weierstrass's form
• Lame equation in the algebraic form