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Mehler-Heine asymptotics for multiple orthogonal polynomials


Author: Walter Van Assche
Journal: Proc. Amer. Math. Soc. 145 (2017), 303-314
MSC (2010): Primary 33C45, 42C05
DOI: https://doi.org/10.1090/proc/13214
Published electronically: July 12, 2016
MathSciNet review: 3565381
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Abstract: Mehler-Heine asymptotics describe the behavior of orthogonal
polynomials near the edges of the interval where the orthogonality measure is supported. For Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the hard edge involves Bessel functions $ J_\alpha $. We show that the asymptotic behavior near the endpoint of the interval of (one of) the measures for multiple orthogonal polynomials involves a generalization of the Bessel function. The multiple orthogonal polynomials considered are Jacobi-Angelesco polynomials, Jacobi-Piñeiro polynomials, multiple Laguerre polynomials, multiple orthogonal polynomials associated with modified Bessel functions (of the first and second kind), and multiple orthogonal polynomials associated with Meijer $ G$-functions.


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Additional Information

Walter Van Assche
Affiliation: Department of Mathematics, KU Leuven, Celestijnenlaan 200B, Box 2400, BE 3001 Leuven, Belgium
Email: walter@wis.kuleuven.be

DOI: https://doi.org/10.1090/proc/13214
Keywords: Multiple orthogonal polynomials, Mehler-Heine asymptotics
Received by editor(s): August 26, 2014
Received by editor(s) in revised form: March 20, 2016, and March 24, 2016
Published electronically: July 12, 2016
Additional Notes: This research was supported by KU Leuven Research Grant OT/12/073, FWO Research Grant G.0934.13 and the Belgian Interuniversity Attraction Poles Programme P7/18.
Communicated by: Mourad Ismail
Article copyright: © Copyright 2016 American Mathematical Society