The Asymptotics of the Mittag–leffler Polynomials

We investigate the asymptotic behaviour of the Mittag-Leffler polynomials Gn(z) for large n and z , where z is a complex variable satisfying 0 arg z 2 π . A summary of the asymptotic properties of Gn(ix) for real values of x and an approximation for its extreme zeros as n→∞ are given. When the variables are such that z/n is finite, an expansion is obtained using the method of steepest descents applied to a suitable integral representation. This expansion holds everywhere in the first quadrant of the z -plane except in the neighbourhood of the point z = in , where there is a coalescence of saddle points. Numerical results are presented to illustrate the accuracy of the various expansions. Mathematics subject classification (2010): Primary 41A60, 33C45.