On the roots of orthogonal polynomials and Euler-Frobenius polynomials
Consider a family of polynomials recursively defined by \(P_0 (x) = x^l\) \((l\) any nonnegative integer) and \[ c_{n + 1} P_{n + 1} (x) = - 2r_n xP_n (x) + (1 - x^2) P_n' (x), \quad n \geq 0, \] where \(r_n > 0\) and \(c_n \in \mathbb{R}\backslash\{0\}\). For example, ultraspherical and Euler-Frobenius polynomials verify this scheme. The authors show that the roots of \(P_n\) are real, simple and belong to \((- 1,1)\). Interlacing properties of the zeros of \(P_n\) and \(P_{n + 1}\) (and \(P_n\) and \(P_{n + 2})\) are established. The method of proof is extended to the study of Hermite like polynomials given by \(H_0 (x) = x^l\) and \[ c_{n + 1} H_{n + 1} (x) = - 2xH_n (x) + H_n'(x), \] where \(c_n \in \mathbb{R}\backslash\{0\}\).
scientific article; zbMATH DE number 861806