### Abstract

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 12
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.

Original language | English |
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Pages (from-to) | 1-16 |

Number of pages | 16 |

Journal | Journal of Classical Analysis |

Volume | 1 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2012 |

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## Cite this

Paris, R. B. (2012). The asymptotics of the mittag-leffler polynomials.

*Journal of Classical Analysis*,*1*(1), 1-16. https://doi.org/10.7153/jca-01-01