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Approximations of orthogonal polynomials in terms of Hermite polynomials. (English) Zbl 0958.33004
It is known that the asymptotics of Laguerre polynomials $L_n^\alpha(x)$ (or Gegenbauer polynomials $C_n^\gamma(x)$) can be expressed in terms of Hermite polynomials for large values of the order parameter $\alpha$ (or $\gamma$). This paper gives a uniform approach, based on generating functions, to derive such asymptotic expressions, not only for these but also for other classes of orhogonal polynomials. The details are worked out for Gegenbauer, Laguerre, Jacobi and Tricomi-Carlitz polynomials. From these asymptotics, estimates for the zeros of the polynomials can be obtained in terms of the zeros of Hermite polynomials. Also this aspect is worked out for the above mentioned classes.

33C45Orthogonal polynomials and functions of hypergeometric type
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
30C15Zeros of polynomials, etc. (one complex variable)
41A10Approximation by polynomials