The analytic solution for the power series expansion of Heun function

• Power series expansion for infinite series of Heun function using 3 term rec. form.
• Power series for polynomial which makes BnBn term terminated of Heun function.
• Applicable to areas such as the Teukolsky equation in Kerr–Newman–de Sitter geometries.

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2F12F1, 1F11F1 and 0F10F1 functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins.In this paper I will apply three term recurrence formula (Y.S. Choun, arXiv:1303.0806, 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of AnAn’s. Section  3 contains my analysis on applying the power series expansions of Heun function to a recent paper (R.S. Maier, Math. Comp. 33 (2007) 811–843). 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  4 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.