Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Mobile Device Pairing
 Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Orthogonal polynomials with a resolvent-type generating function

Author(s): Michael Anshelevich
Journal: Trans. Amer. Math. Soc. 360 (2008), 4125-4143.
MSC (2000): Primary 05E35; Secondary 46L54, 33C47
Posted: February 27, 2008
MathSciNet review: 2395166
Retrieve article in: PDF

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:

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)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 05E35, 46L54, 33C47

Retrieve articles in all Journals with MSC (2000): 05E35, 46L54, 33C47

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
Email: manshel@math.tamu.edu

DOI: 10.1090/S0002-9947-08-04368-7
PII: S 0002-9947(08)04368-7
Received by editor(s): October 25, 2004
Received by editor(s) in revised form: June 7, 2006
Posted: February 27, 2008
Additional Notes: This work was supported in part by an NSF grant DMS-0400860
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia