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

Original · EN

We consider the differential equations y''=λ₀(x)y'+s₀(x)y, where λ₀(x), s₀(x) are C∞-functions. We prove (i) if the differential equation, has a polynomial solution of degree n >0, then δₙ=λₙ sₙ₋₁-λₙ₋₁sₙ=0, where λₙ= λₙ₋₁′+sₙ₋₁+λ₀λₙ₋₁and sₙ=sₙ₋₁′+s₀λₖ₋₁, n=1,2,.... Conversely (ii) if λₙλₙ₋₁≠ 0 and δₙ=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.