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Monic non-commutative orthogonal polynomials


Author: Michael Anshelevich
Journal: Proc. Amer. Math. Soc. 136 (2008), 2395-2405
MSC (2000): Primary 05E35; Secondary 46Nxx
DOI: https://doi.org/10.1090/S0002-9939-08-09306-4
Published electronically: February 20, 2008
MathSciNet review: 2390506
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Abstract: Among all states on the algebra of non-commutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankel-type determinants, and a representation as a joint distribution of operators on a Fock space.


References [Enhancements On Off] (What's this?)

  • [Ans06] Michael Anshelevich, Orthogonal polynomials with a resolvent-type generating function, math.CO/0410482, accepted for publication by Transactions of the AMS, 2006.
  • [Ans07] -, Free Meixner states, math.CO/0702158, accepted for publication by Communications in Mathematical Physics, 2007.
  • [BC04] T. Banks and T. Constantinescu, Orthogonal polynomials in several non-commuting variables. II, math.FA/0412528, 2004.
  • [BCJ05] T. Banks, T. Constantinescu, and J. L. Johnson, Relations on non-commutative variables and associated orthogonal polynomials, Operator theory, systems theory and scattering theory: multidimensional generalizations, Oper. Theory Adv. Appl., vol. 157, Birkhäuser, Basel, 2005, pp. 61–90. MR 2129643, https://doi.org/10.1007/3-7643-7303-2_2
  • [DES] Ioana Dumitriu, Alan Edelman, and Gene Shuman, MOPS: multivariate orthogonal polynomials (symbolically), J. Symbolic Comput. 42 (2007), no. 6, 587–620. MR 2325917, https://doi.org/10.1016/j.jsc.2007.01.005
  • [DX01] Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 2002m:33001
  • [Fla80] P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980), no. 2, 125-161. MR 82f:05002a

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Additional Information

Michael Anshelevich
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: manshel@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09306-4
Received by editor(s): February 8, 2007
Received by editor(s) in revised form: June 19, 2007
Published electronically: February 20, 2008
Additional Notes: This work was supported in part by NSF grant DMS-0613195
Communicated by: Jim Haglund
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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