ABSTRACT
This chapter discusses asymptotic forms of Cnγ(x), Lnα(x), Pn(α, β)(x), the Gegenbauer, Laguerre and Jacobi polynomials. The asymptotic behavior of these classical orthogonal polynomials has been the subject of several investigations. The Laguerre polynomial is in fact a special case of the Whittaker, or confluent hypergeometric function. The Jacobi polynomial and the Gegenbauer polynomial are special cases of the Gauss hypergeometric function, for which class rather few interesting uniform expansions have been investigated. The estimates of the orthogonal polynomials are obtained by using differential equations, and Liouville Green transformations thereof. The chapter demonstrates the usefulness of the estimates by comparing the zeros of the Jacobi polynomials with transformed zeros of the LG approximants for these cases: the Gegenbauer polynomials and the Laguerre polynomials.
