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

Let {Pn(x)}n=0∞ be a system of polynomials satisfying the recurrence relation P−1(x) = 0, P0(x) = 1, Pn+1(x) + hnPn−1(x) + cnPn(x) = xPn(x), where hn, cn are real sequences and hn > 0, n = 0, 1, 2, …. The co-recursive polynomials {Pn∗(x)}n=0∞ satisfy the same recurrence relation except for n = 1, where P1∗(x) = γx − c0 − β, γ ≠ 0. It is well known that the problem of determining the zeros of Pn(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, λ1, λ2, …, λk) is found, k = 1, 2, …, n, whose zeros are the zeros of Pn∗(x), and λk are the zeros of the polynomial Pn(x) of degree n, for γ ≠ 0. This function unifies many results concerning interlacing between the zeros of Pn(x) and Pn∗(x) for γ ≠ 0. Moreover we obtain from this function similar results in the unstudied case γ = 0.