Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions

Summary.

Let w _α,beta (t)=(1−t)^α(1+t)^β, α,β>−1, denote the Jacobi weight function. For 0≤α,β<5/2 it is shown that on compact subintervals of (−1,1), the corresponding Stieltjes polynomials and their derivatives are asymptotically equal to certain Jacobi polynomials. This also leads to asymptotic representations of those weights of Gauss-Kronrod quadrature formulae, which correspond to nodes in a compact subinterval of (−1,1). On the other hand, it is demonstrated that for the parameters satisfying min(α,β)≥0 and max(α,β)>5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions ${{w_\lambda(t) = \big(1-t^2\big)^{{\lambda-1/2}}}}$ for 0≤λ≤2. For λ>3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently.