Near-Legendre Differential Equations

A differential equation of the form ((1-x^2m ) y^((k)) )^((2m-k))+λy=0,-1≤x≤1,0≤k≤2m;k,m integers is called a near-Legendre equation. We show that such an equation has infinitely many polynomial solutions corresponding to infinitely many λ. We list all of these equations for 1≤m≤2. We show, for m=1, that these solutions are 'partially' orthogonal with respect to some weight functions and show how to expand functions using these polynomials. We give few applications to partial differential equations.