ON THE ROOTS OF ORTHOGONAL POLYNOMIALS AND EULER-FROBENIUS POLYNOMIALS

Interlacing properties of the roots of the polynomials P-n(x) and P-n+1(x) and P-n(x) and P-n+2(x) are obtained for sequences of polynomials generated recursively by the scheme: P-0(x) = x(l) (l a nonnegative integer) and c(n+1)/P-n+1(x) = -2r(n)xP(n)(x) + (1 - x(2))DPn(x). Ultraspherical polynomials and Euler-Frobenius polynomials are examples of such sequences. We obtain similar results for Hermite like polynomials obtained by the scheme: H-0(x) = x(l) (l a nonnegative integer) and H-n+1(x) = -2xH(n)(x) + DHn(x). (C) 1995 Academic Press, Inc.