The asymptotics of the mittag-leffler polynomials

Richard B. Paris

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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 languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Classical Analysis
Volume1
Issue number1
DOIs
Publication statusPublished - 2012

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Polynomial
Steepest Descent
Quadrant
Coalescence
Complex Variables
Saddlepoint
Integral Representation
Asymptotic Properties
Extremes
Asymptotic Behavior
Numerical Results
Zero
Approximation

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Paris, Richard B. / The asymptotics of the mittag-leffler polynomials. In: Journal of Classical Analysis. 2012 ; Vol. 1, No. 1. pp. 1-16.
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Paris, RB 2012, 'The asymptotics of the mittag-leffler polynomials', Journal of Classical Analysis, vol. 1, no. 1, pp. 1-16. https://doi.org/10.7153/jca-01-01

The asymptotics of the mittag-leffler polynomials. / Paris, Richard B.

In: Journal of Classical Analysis, Vol. 1, No. 1, 2012, p. 1-16.

Research output: Contribution to journal › Article

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N2 - 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.

AB - 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.

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Paris RB. The asymptotics of the mittag-leffler polynomials. Journal of Classical Analysis. 2012;1(1):1-16. https://doi.org/10.7153/jca-01-01

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