Perturbation of the coefficients in the recurrence relation of a class of polynomials

Abstract

Let { P n ( x)} n=0 ∞ be a system of polynomials satisfying the recurrence relation P −1( x) = 0, P 0( x) = 1, P n+1 ( x) + h n P n−1 ( x) + c n P n ( x) = xP n ( x), where h n , c n are real sequences and h n > 0, n = 0, 1, 2, …. The co-recursive polynomials {P n ∗(x)} n=0 ∞ satisfy the same recurrence relation except for n = 1, where P 1 ∗(x) = γx − c 0 − β , γ ≠ 0. It is well known that the problem of determining the zeros of P n ( x) is equivalent to the problem of determining the eigenvalues of a generalized eigenvalue problem Tƒ = λAƒ, where T and A are symmetric matrices. In this paper the problem of determining the zeros of the co-recursive polynomials is reduced to a perturbation problem of the operators T and A perturbed by perturbations of rank one. A function ϕ( λ) = ϕ( λ, λ 1, λ 2, …, λ k ) is found, k = 1, 2, …, n, whose zeros are the zeros of P n ∗(x) , and λ k are the zeros of the polynomial P n ( x) of degree n, for γ ≠ 0. This function unifies many results concerning interlacing between the zeros of P n ( x) and P n ∗(x) for γ ≠ 0. Moreover we obtain from this function similar results in the unstudied case γ = 0.