Criterion for polynomial solutions to a class of linear differential equations of second order

摘要

We consider the differential equations y" = lambda(0)( x) y' + s(0)(x) y, where lambda(0)(x), s(0)(x) are C-infinity-functions. We prove ( i) if the differential equation has a polynomial solution of degree n > 0, then delta(n) = lambda(n)s(n-1) -lambda(n-1)s(n) = 0, where lambda(n) =lambda'(n-1) + s(n-1) + lambda(0)lambda(n-1) and s(n) = s'(n-1) + s(0)lambda(k-1), n = 1, 2,.... Conversely (ii) if lambda(n)lambda(n-1) not equal 0 and delta(n) = 0, then the differential equation has a polynomial solution of degree at most n. We show that the classical differential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev ( first and second kinds), Gegenbauer and the Hypergeometric type, etc obey this criterion. Further, we find the polynomial solutions for the generalized Hermite, Laguerre, Legendre and Chebyshev differential equations.