Characterizations of orthogonal polynomials satisfying differential equations

Abstract

We consider the characterizations of orthogonal polynomials satisfying a differential equation of the form $L_N(y)=∑_{i=0}^Nl_i(x)y^i(x)= λy$. In 1923, S.Bochner classified all polynomial solutions of the differential equation with N=2. He showed that up to a complex linear change of variable, the only polynomial systems that arise as eigen solutions of the differential equation are Jacobi polynomials, Laguerre polynomials, hermite polynomials, ${X^n}_{n=0}^∞$, and Bessel polynomials. It is easy to see that the polynomial system ${X^n }_{n=0}^∞$ can not be orthogonal. The orthogonality of Jacobi, Laguerre, Hermite, and Bessel Polynomials are precisely investigated by many authors. They are nowcalled classical orthogonal polynomials. In 1935, W. Hahn proved that the only orthogonal polynomials whose derivatives also form an orthogonal polynomial system are classical orthogonal polynomials. He later extend his result by rooving that if ${P_{n(x)}}_{n=0}^∞$ is an orthogonal polynomial system such that ${P_n^{(r)}}_{n=0}^∞$ is also an orthogonal polynomial system for some integer r ≥ 1, then the polynomial system ${P_{n(x)}}_{n=0}^∞$ must be a classical orthogonal polynomial. Besides Bochner``s and Hahn``s, there are many other properties common to all of clasical orthogonal polynomials. In chapter two, we give simple proofs of known characterization theorems and new characterization of classical orthogonal polynomials which improves Hahn``s theorem. For N>2, H.L.Krall found a remarkable theorem characterizing all differential equations of order N, which have orthogonal polynomials as solutions. In chapter three, we give new and simple proof of Krall``s. In chapter four, as an application of this characterization, we ger new characterizatio s of classical orthogonal polynomials which generalize Bochner``s and Hahn``s theorems.