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 Bn term terminated including all higher terms of An's. In this series I will show how to obtain the formula for the polynomial which makes An term terminated including all higher terms of Bn's and infinite series of its power series expansion. In the future series I will show you for the polynomial which makes An and Bn 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 An term terminated including all higher terms of Bn'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 An term terminated.