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Quadratic harnesses, $ q$-commutations, and orthogonal martingale polynomials


Authors: Wlodzimierz Bryc, Wojciech Matysiak and Jacek Wesolowski
Journal: Trans. Amer. Math. Soc. 359 (2007), 5449-5483
MSC (2000): Primary 60J25; Secondary 46L53
DOI: https://doi.org/10.1090/S0002-9947-07-04194-3
Published electronically: June 13, 2007
MathSciNet review: 2327037
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Abstract: We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a $ q$-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.


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Additional Information

Wlodzimierz Bryc
Affiliation: Department of Mathematics, University of Cincinnati, P.O. Box 210025, Cincinnati, Ohio 45221–0025
Email: Wlodzimierz.Bryc@UC.edu

Wojciech Matysiak
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
Email: matysiak@mini.pw.edu.pl

Jacek Wesolowski
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warszawa, Poland
Email: wesolo@alpha.mini.pw.edu.pl

DOI: https://doi.org/10.1090/S0002-9947-07-04194-3
Keywords: Quadratic conditional variances, harnesses, orthogonal martingale polynomials, hypergeometric orthogonal polynomials
Received by editor(s): June 8, 2005
Received by editor(s) in revised form: September 26, 2005
Published electronically: June 13, 2007
Additional Notes: This research was partially supported by NSF grants #INT-0332062, #DMS-0504198, and by the C.P. Taft Memorial Fund.
Article copyright: © Copyright 2007 American Mathematical Society

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