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Transactions of the Moscow Mathematical Society
Transactions of the Moscow Mathematical Society
ISSN 1547-738X(online) ISSN 0077-1554(print)

 

Asymptotics of Meixner polynomials and Christoffel-Darboux kernels


Authors: A. I. Aptekarev and D. N. Tulyakov
Translated by: V. E. Nazaikinskii
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva.
Journal: Trans. Moscow Math. Soc. 2012, 67-106
MSC (2010): Primary 33C45; Secondary 60B10
Published electronically: January 24, 2013
MathSciNet review: 3184968
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Abstract: We obtain the asymptotics of the classical Meixner polynomials (orthogonal with respect to a discrete measure supported at the nonnegative integer points) and the corresponding reproducing kernels (Christoffel-Darboux kernels) as the number $ n$ of the polynomial and the variable $ x$ tend to infinity under various relationships between their growth rates. (These asymptotics are known as the Plancherel-Rotach asymptotics.)


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Additional Information

A. I. Aptekarev
Affiliation: Keldysh Institute of Applied Mathematics, 4 Miusskaya pl., 125047 Moscow, Russian Federation
Email: aptekaa@keldysh.ru

D. N. Tulyakov
Affiliation: Keldysh Institute of Applied Mathematics, 4 Miusskaya pl., 125047 Moscow, Russian Federation
Email: dntulyakov@gmail.com

DOI: http://dx.doi.org/10.1090/S0077-1554-2013-00203-4
PII: S 0077-1554(2013)00203-4
Keywords: Discrete orthogonal polynomials, point determinant process, asymptotic method, Christoffel–Darboux kernel
Published electronically: January 24, 2013
Additional Notes: This work was supported in part by RFBR grants no. 11-01-00245 and 11-01-12045-ofi-m and by Program no. 1 of the Branch of Mathematics, Russian Academy of Sciences. The first author was also supported by the Program “Cátedras de Excelencia” of the Universidad Carlos III, Madrid, and Banco Santander. The second author was also supported by RFBR grant No. 10-01-000682.
Dedicated: Dedicated to A. A. Gonchar on the occasion of his eightieth birthday
Article copyright: © Copyright 2013 American Mathematical Society