The analytic solution for the power series expansion of Heun function

Abstract

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric 2 F 1 , 1 F 1 and 0 F 1 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 (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. 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.

Closed-form solution of a general three-term recurrence relation: applications to Heun functions and social choice models

“…It elucidates the functional dependence of C i on the coefficients P j , Q j and R j in the recurrence relation itself. Note that this result is much more compact, and its derivation much easier, than other solutions to three-term recurrence relations described in [14,17].…”

“…Although commonly stated as being unsolved (e.g., see the section on Heun functions in the handbook of Maple [13]), some papers in the past decade have made progress in solving three-term recurrence relations. Recent work by Choun [14], from a series of studies that include [15,16], tackles the problem of solving the three-term recurrence relation defining the Heun function and looks to find the conditions under which Heun functions reduce to finite polynomials. Unfortunately, the proposed solution is difficult to verify and unwieldy (see [14, Eq.…”

Isomonodromy, Painlevé transcendents and scattering off of black holes

“…and for the singularity at z = ∞, 24) where the last equality follows from the invariance of θ under homographic transformations. It is possible to show that θ ∞ does not depend on C ℓ .…”

“…First, the coefficients of the series solution obey a threeterm recurrence relation [23], which is not easily tractable to find explicit solutions [24]. Second, there is no known integral representation of Heun functions in terms of elementary functions, which hinders a direct treatment of the monodromies.…”

Exact solution of two-thirds power potential

“…Eigenfunctions and eigenvalues. -To construct the solution, we first express the regular solution (13) as a linear combination of the two irregular solutions (14) and (15).…”