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New inequalities for the zeros of Jacobi polynomials. (English) Zbl 0639.33012

The author makes ingenious use of the Sturm comparison theorem to provide upper and lower bounds for the zeros of the Jacobi polynomial P n (α,β) (cosθ), is case -α,β. He shows that an asymptotic formula, involving zeros of Bessel functions, due to Frenzen and Wong, in fact provides a lower bound for these zeros (and also an upper bound, using P n (α,β) (x)=(-1) n P n (β,α) (-x))· He also shows that between any pair of zeros there occurs at least one root of a certain transcendental equation involving elementary functions. In the case of the kth zero, θ n,k (α), k=1,2,···,[n/2], of the ultraspherical polynomial P n (α,α) (cosθ), this leads to the inequalities

ϕ n,k (α)θ n,k (α)ϕ n,k (α)+N -2 ((1/8)-α 2 /2)cotϕ nk (α),

where N=n+α+ and ϕ n,k (α)=(k+α/2-1/4)π/N. Comparisons are made with known bounds and numerical examples are given to illustrate the sharpness of the new inequalities.

Reviewer: M.E.Muldoon

MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
65D20Computation of special functions, construction of tables