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

Abstract

We consider the differential equations y''=\\lambda_0(x)y'+s_0(x)y, where\n\\lambda_0(x), s_0(x) are C^{\\infty}-functions. We prove (i) if the differential\nequation, has a polynomial solution of degree n >0, then \\delta_n=\\lambda_n\ns_{n-1}-\\lambda_{n-1}s_n=0, where \\lambda_{n}=\n\\lambda_{n-1}^\\prime+s_{n-1}+\\lambda_0\\lambda_{n-1}\\hbox{and}\\quad\ns_{n}=s_{n-1}^\\prime+s_0\\lambda_{k-1},\\quad n=1,2,.... Conversely (ii) if\n\\lambda_n\\lambda_{n-1}\\ne 0 and \\delta_n=0, then the differential equation has\na polynomial solution of degree at most n. We show that the classical\ndifferential equations of Laguerre, Hermite, Legendre, Jacobi, Chebyshev (first\nand second kind), Gegenbauer, and the Hypergeometric type, etc, obey this\ncriterion. Further, we find the polynomial solutions for the generalized\nHermite, Laguerre, Legendre and Chebyshev differential equations.\n