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Orthogonal polynomials with a resolvent-type generating function

Author: Michael Anshelevich
Journal: Trans. Amer. Math. Soc. 360 (2008), 4125-4143
MSC (2000): Primary 05E35; Secondary 46L54, 33C47
Published electronically: February 27, 2008
MathSciNet review: 2395166
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Abstract | References | Similar Articles | Additional Information

Abstract: The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favard-type recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. Their recursion relations have a more special form; the best way to describe them is in terms of the free cumulant generating function of the state of orthogonality, which turns out to satisfy a type of second-order difference equation. If the difference equation is in fact first order, and the state is tracial, we show that the state is necessarily a rotation of a free product state. We also describe interesting examples of non-tracial infinitely divisible states with orthogonal free Sheffer polynomials.

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

  • 1. AL-SALAM, W. A. and CHIHARA, T. S. (1987).
    $ q$-Pollaczek polynomials and a conjecture of Andrews and Askey.
    SIAM J. Math. Anal. 18 228-242. MR 871834 (88a:33019)
  • 2. ANSHELEVICH, M. (2001).
    Partition-dependent stochastic measures and $ q$-deformed cumulants.
    Doc. Math. 6 343-384 (electronic). MR 1871667 (2004k:46107)
  • 3. ANSHELEVICH, M. (2003).
    Free martingale polynomials.
    J. Funct. Anal. 201 228-261. MR 1986160 (2004f:46079)
  • 4. ANSHELEVICH, M. (2004).
    Appell polynomials and their relatives.
    Int. Math. Res. Not. n.65, 3469-3531. MR 2101359 (2005k:33012)
  • 5. ASKEY, R. and WILSON, J. (1985).
    Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials.
    Mem. Amer. Math. Soc. 54 iv+55. MR 783216 (87a:05023)
  • 6. BANKS, T., CONSTANTINESCU, T. and JOHNSON, J. L. (2005).
    Relations on non-commutative variables and associated orthogonal polynomials.
    In Operator theory, systems theory and scattering theory: multidimensional generalizations, vol. 157 of Oper. Theory Adv. Appl. Birkhäuser, Basel, 61-90. MR 2129643 (2006d:47012)
  • 7. BOŻEJKO, M. and BRYC, W. (2006).
    On a class of free Lévy laws related to a regression problem.
    J. Funct. Anal. 236 59-77. MR 2227129 (2007a:46071)
  • 8. CASALIS, M. (1991).
    Les familles exponentielles à variance quadratique homogène sont des lois de Wishart sur un cône symétrique.
    C. R. Acad. Sci. Paris Sér. I Math. 312 537-540. MR 1099688 (91m:60029)
  • 9. CASALIS, M. (1996).
    The $ 2d+4$ simple quadratic natural exponential families on $ {\bf R}\sp d$.
    Ann. Statist. 24 1828-1854. MR 1416663 (97h:60011)
  • 10. DUNKL, C. F. and XU, Y. (2001).
    Orthogonal polynomials of several variables, vol. 81 of Encyclopedia of Mathematics and its Applications.
    Cambridge University Press, Cambridge. MR 1827871 (2002m:33001)
  • 11. FEINSILVER, P. (1986).
    Some classes of orthogonal polynomials associated with martingales.
    Proc. Amer. Math. Soc. 98 298-302. MR 854037 (88d:60136)
  • 12. GLOCKNER, P., SCHüRMANN, M. and SPEICHER, R. (1992).
    Realization of free white noises.
    Arch. Math. (Basel) 58 407-416. MR 1152632 (93e:46075)
  • 13. HADWIN, D. (2001).
    A noncommutative moment problem.
    Proc. Amer. Math. Soc. 129 1785-1791 (electronic). MR 1814111 (2003a:46101)
  • 14. LETAC, G. (1989).
    Le problème de la classification des familles exponentielles naturelles de $ {\bf R}\sp d$ ayant une fonction variance quadratique.
    In Probability measures on groups, IX (Oberwolfach, 1988), vol. 1379 of Lecture Notes in Math. Springer, Berlin, 192-216. MR 1020532 (92a:60041)
  • 15. MEIXNER, J. (1934).
    Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion.
    J. London Math. Soc. 9 6-13.
  • 16. NICA, A. (1999).
    $ R$-transforms in free probability.
    Lecture notes for an IHP course (unpublished).
  • 17. POMMERET, D. (1996).
    Orthogonal polynomials and natural exponential families.
    Test 5 77-111. MR 1410457 (97j:62016)
  • 18. POMMERET, D. (2000).
    Orthogonality of the Sheffer system associated to a Levy process.
    J. Statist. Plann. Inference 86 1-10. MR 1763177 (2003b:60063)
  • 19. SPEICHER, R. (1997).
    Free probability theory and non-crossing partitions.
    Sém. Lothar. Combin. 39 Art. B39c, 38 pp. (electronic). MR 1490288 (98m:46081)
  • 20. VOICULESCU, D. (2000).
    Lectures on free probability theory.
    In Lectures on probability theory and statistics (Saint-Flour, 1998), vol. 1738 of Lecture Notes in Math. Springer, Berlin, 279-349. MR 1775641 (2001g:46121)

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Additional Information

Michael Anshelevich
Affiliation: Department of Mathematics, University of California, Riverside, California 92521-0135
Address at time of publication: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): October 25, 2004
Received by editor(s) in revised form: June 7, 2006
Published electronically: February 27, 2008
Additional Notes: This work was supported in part by an NSF grant DMS-0400860
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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