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

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.

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

References in corpus (9)

• Heun equation, Teukolsky equation, and type-D metrics
• The generalized Heun equation in QFT in curved space-times
• Classes of Exact Solutions to Regge-Wheeler and Teukolsky Equations
• Generalization of the three-term recurrence formula and its applications
• Asymptotic behavior of Heun function and its integral formalism
• Approximative solution of the spin free Hamiltonian involving only scalar potential for the quark-antiquark system
• The integral formalism and the generating function of grand confluent hypergeometric function
• Analytic solution for grand confluent hypergeometric function
• On complex oscillation theory, quasi-exact solvability and Fredholm Integral Equations

Cited by in corpus (4)

• Asymptotic behavior of Heun function and its integral formalism
• Analytic solution for grand confluent hypergeometric function
• Complete polynomials using 3-term and reversible 3-term recurrence formulas (3TRF and R3TRF)
• The violation of a uniqueness theorem and an invariant in the application of Poincaré--Perron theorem to Heun's equation