Asymptotic approximations of Jacobi polynomials and their zeros with error bounds

Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P
n
^{(α, β)}
(cose θ) (0 ≤ θ ≤ π/2,α,β >−1), as n →+ ∞. An accurate approximation with error bounds is also constructed for the zero θ
n, s
of P
n
^{(α, β)}
(cosθ) (α ≥ 0, β > − 1).