The analytic solution for the power series expansion of Heun function

TL;DR

This paper derives an analytic power series expansion for the Heun function using a three-term recurrence formula, providing explicit solutions for all higher terms and applications to various special functions.

Contribution

It introduces a closed-form power series expansion for the Heun function utilizing a three-term recurrence formula, extending previous methods and including all higher terms.

Findings

Derived explicit power series expansion for Heun functions

Applied the expansion to analyze solutions in mathematical physics

Provided examples demonstrating the utility of the series expansion

Abstract

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric_2F_1,_1F_1 and_0F_1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrodinger equation of quantum mechanics, and addition of three quantum spins. In this paper I will apply three term recurrence formula (Choun, Y.S., arXiv:1303.0806., 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of A_n's. Section three contains my analysis on applying the power series expansions of Heun function to a recent paper. (R.S. Maier, Math. Comp. 33, 2007) Due to space restriction final equations for the 192 Heun functions are not included in the paper, but feel free to contact me for the final solutions. Section four contains two additional examples using the power series expansions of Heun function. This paper is 3rd out of 10 in series "Special functions and three term recurrence formula (3TRF)". See section 5 for all the papers in the series. The previous paper in series deals with three term recurrence formula (3TRF). The next paper in the series describes the integral forms of Heun function and its asymptotic behaviors analytically.