On generalized Heun equation with some mathematical properties

Anotace: We study the analytic solutions of the generalized Heun equation, (α0 + α1r + α2r2 + α3r3) y′′ + (β0 + β1r + β2r2) y′ + (ε0 + ε1r) y = 0, where |α3| + |β2|≠ 0, and {αi}3i=0, {βi}2i=0, {εi}1i=0 are real parameters. The existence conditions for the polynomial solutions are given. A simple procedure based on a recurrence relation is introduced to evaluate these polynomial solutions explicitly. For α0 = 0, α1≠ 0, we prove that the polynomial solutions of the corresponding differential equation are sources of finite sequences of orthogonal polynomials. Several mathematical properties, such as the recurrence relation, Christoffel-Darboux formulas and the norms of these polynomials, are discussed. We shall also show that they exhibit a factorization property that permits the construction of other infinite sequences of orthogonal polynomials.