Factorization of the Heun's differential operator

The differential equation with four regular singularities located at z = 0, 1, a and infinity, called the Heun's equation (HE), is y"(z) + [ gamma/z + delta/z - 1 + epsilon/z - a]y'(z) + alphabetaz - q/z(z - 1)(z - a) y(z) = 0 with alpha + beta + 1 = gamma + delta + epsilon, and defines the Heun's operator H by H[y(z)] = {P-3(z)D-2 +P-2(z)D +P-1(z)}[y(z)] with D equivalent to d/dz and P-i(z) polynomials of degree i. H can be factorized in the form H = [L(z)D + M(z)][(L) over bar (z)D + (M) over bar (z)] Polynomials L, (L) over bar, M and (M) over bar are given explicitly in the cases where this factorization is possible. It is shown that the value of the parameters alpha, beta and q allowing the factorization coincides with those obtained from the F-homotopic transformation: y(z) = z(rho)(z - 1)(sigma)(z - a)(tau)(y) over tilde (z) forcing (y) over tilde (z) to be solution of a HE as y(z). (C) 2002 Elsevier Science Inc. All rights reserved.