ON GENERALIZED HEUN EQUATION WITH SOME MATHEMATICAL PROPERTIES

We study the analytic solutions of the generalized Heun equation, (alpha(0) + alpha(1) r + alpha(2) r(2) + alpha(3) r(3)) y '' + (beta(0) + beta(1) r + beta(2) r(2)) y' + (epsilon(0) + epsilon(1) r) y = 0, where vertical bar alpha(3)vertical bar + vertical bar beta(2)vertical bar not equal 0, and {alpha(i)}(i=0)(3), {beta(i)}(i=0)(2),{epsilon(i)}(i=0)(1) 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 alpha(0) = 0, alpha(1) not equal 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.