A characterization of Hermite polynomials

Abstract

We show that for any orthogonal polynomials {R(n)(x)}(infinity)(n=0) satisfying a spectral type differential equation of order N (greater than or equal to 2) L(N)[y](x) = (i=1)Sigma(N) l(i)(x)y((i))(x) = lambda(n)y(x), {P-n(x)}(infinity)(n=0) must be essentially Hermite polynomials if and only if the leading coefficient l(N)(x) is a nonzero constant.