Asymptotic approximations of Jacobi functions and their zeros with error bounds

The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variable x(t), we've got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions partial derivative((alpha,beta))(mu) (t) (alpha > -1) as mu -> + infinity. Besides, an accurate approximation with error bounds is also constructed correspondingly for the zeros t(mu,s) of partial derivative((alpha,beta))(mu) (t) (alpha >= 0) as mu -> + infinity, uniformly with respect to s = 1, 2,....