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

Abstract

We consider the differential equations y''=λ_0(x)y'+s_0(x)y, where λ_0(x), s_0(x) are C^{\infty}-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then δ_n=λ_n s_{n-1}-λ_{n-1}s_n=0, where λ_{n}= λ_{n-1}^\prime+s_{n-1}+λ_0λ_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0λ_{k-1},\quad n=1,2,.... Conversely (ii) if λ_nλ_{n-1}\ne 0 and δ_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 kind), 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.