Polynomial Asymptotic Estimates of Gegenbauer, Laguerre, and Jacobi Polynomials

Abstract

We discuss asymptotic forms of CJ( x ), L~( x), P~0t,/3) ( x ), the Gegenbauer, Laguerre and Jacobi polynomials. The asymptotic behavior of these classical orthogonal polynomials has been the subject of several investigations. The research usually concentrates on the case that the degree n of the polynomial is the large parameter, and for all classical orthogonal polynomials the asymptotic behavior is well established now. Inside the domain of the zeros of the polynomial the behavior can be described in terms of elementary functions, such as trigonometric functions. In the domain where the transition from oscillatory to monotonic behavior occurs, familiar higher transcendental functions can be used as estimates. For example, the "first" zeros of the Jacobi polynomial can be approximated in terms of the zeros of the J Bessel function. In SzEGO ( 1958) several classical results can be found. For Jacobi polynomials he gives an estimate of