On the Roots of Orthogonal Polynomials and Euler-Frobenius Polynomials

Interlacing properties of the roots of the polynomials Pn(x) and Pn+1(x) and Pn(x) and Pn+2(x) are obtained for sequences of polynomials generated recursively by the scheme: P0(x) = xl (l a nonnegative integer) and cn+1Pn+1(x) = −2rnxPn(x) + (1 − x2)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: H0(x) = xl (l a nonnegative integer) and Hn+1(x) = −2xHn(x) + DHn(x).