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

TL;DR

This paper develops a generalized framework for three-term recurrence relations in linear differential equations, enabling new analytical solutions for special functions like Heun, Mathieu, Lame, and GCH equations.

Contribution

It introduces a reversible three-term recurrence formula applicable to power series, integrals, and generating functions of key special functions, expanding analytical solution methods.

Findings

Derived closed-form power series expansions

Established integral representations of solutions

Constructed generating functions for special functions

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.