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Asymptotic approximations of Jacobi polynomials are given in terms ofelementary functions for large degree $n$ and parameters $\alpha$ and $\beta$.From these new results, asymptotic expansions of the zeros are derived andmethods are given to obtain the coefficients in the expansions. Theseapproximations can be used as initial values in iterative methods for computingthe nodes of Gauss--Jacobi quadrature for large degree and parameters. Theperformance of  the asymptotic approximations for computing the nodes and weights of theseGaussian quadratures is illustrated with numerical examples.

Asymptotic expansions of Jacobi polynomials and of the nodes and weights  of Gauss-Jacobi quadrature for large degree and parameters in terms of  elementary functions