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

TL;DR

This paper establishes a criterion based on a specific determinant condition for the existence of polynomial solutions to a class of second-order linear differential equations, and applies it to classical special functions.

Contribution

It introduces a new necessary and sufficient condition for polynomial solutions, extending to generalized forms and classical special functions.

Findings

Polynomial solutions exist if nd only if  certain determinant ondition is satisfied.

Classical equations like Laguerre, Hermite, Legendre, and others satisfy this criterion.

Explicit polynomial solutions are derived for generalized Hermite, Laguerre, Legendre, and Chebyshev equations.

Abstract

We consider the differential equations y''=\lambda_0(x)y'+s_0(x)y, where \lambda_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 \delta_n=\lambda_n s_{n-1}-\lambda_{n-1}s_n=0, where \lambda_{n}= \lambda_{n-1}^\prime+s_{n-1}+\lambda_0\lambda_{n-1}\hbox{and}\quad s_{n}=s_{n-1}^\prime+s_0\lambda_{k-1},\quad n=1,2,.... Conversely (ii) if \lambda_n\lambda_{n-1}\ne 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 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.