### Appell polynomials and their relatives III. Conditionally free theory

Michael Anshelevich
Source: Illinois J. Math. Volume 53, Number 1 (2009), 39-66.

#### Abstract

We extend to the multivariate noncommutative context the descriptions of a “once-stripped” probability measure in terms of Jacobi parameters, orthogonal polynomials, and the moment generating function. The corresponding map Φ on states was introduced previously by Belinschi and Nica. We then relate these constructions to the c-free probability theory, which is a version of free probability for algebras with two states, introduced by Bożejko, Leinert, and Speicher. This theory includes the free and Boolean probability theories as extreme cases. The main objects in the paper are the analogs of the Appell polynomial families in the two state context. They arise as fixed points of the transformation which takes a polynomial family to the associated polynomial family (in several variables), and their orthogonality is also related to the map Φ above. In addition, we prove recursions, generating functions, and factorization and martingale properties for these polynomials, and describe the c-free version of the Kailath–Segall polynomials, their combinatorics, and Hilbert space representations.

First Page:
Primary Subjects: 46L53, 46L54, 05E35
We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.
If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Permanent link to this document: http://projecteuclid.org/euclid.ijm/1264170838
Zentralblatt MATH identifier: 05676308
Mathematical Reviews number (MathSciNet): MR2584934

### References

L. Accardi and M. Bożejko, Interacting Fock spaces and Gaussianization of probability measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 663--670.
Mathematical Reviews (MathSciNet): MR1665281
Zentralblatt MATH: 0922.60013
Digital Object Identifier: doi:10.1142/S0219025798000363
N. I. Akhiezer, The classical moment problem and some related questions in analysis, Hafner Publishing Co., New York, 1965; translated by N. Kemmer.
Mathematical Reviews (MathSciNet): MR0184042
Zentralblatt MATH: 0135.33803
M. Anshelevich, Appell polynomials and their relatives, Int. Math. Res. Not. 65 (2004), 3469--3531.
Mathematical Reviews (MathSciNet): MR2101359
Zentralblatt MATH: 1086.33012
Digital Object Identifier: doi:10.1155/S107379280413345X
M. Anshelevich, Free Meixner states, Comm. Math. Phys. 276 (2007), 863--899.
Mathematical Reviews (MathSciNet): MR2350440
Zentralblatt MATH: 1133.33001
Digital Object Identifier: doi:10.1007/s00220-007-0322-3
M. Anshelevich, Monic non-commutative orthogonal polynomials, Proc. Amer. Math. Soc. 136 (2008), 2395--2405.
Mathematical Reviews (MathSciNet): MR2390506
Zentralblatt MATH: 1152.05389
Digital Object Identifier: doi:10.1090/S0002-9939-08-09306-4
M. Anshelevich, Orthogonal polynomials with a resolvent-type generating function, Trans. Amer. Math. Soc. 360 (2008), 4125--4143.
Mathematical Reviews (MathSciNet): MR2395166
Zentralblatt MATH: 1146.05058
Digital Object Identifier: doi:10.1090/S0002-9947-08-04368-7
M. Anshelevich, Appell polynomials and their relatives II. Boolean theory, Indiana Univ. Math. J. 58 (2009), 929--968.
Mathematical Reviews (MathSciNet): MR2514394
Zentralblatt MATH: 1176.46060
Digital Object Identifier: doi:10.1512/iumj.2009.58.3523
M. Anshelevich, Free evolution on algebras with two states, to appear in Reine und Angew. Math., available at arXiv :0803.4280 [math.OA], 2009.
M. P. Appell, Sur une classe de polynômes, Ann. Sci. Ecole Norm. Sup. 9 (1880), 119--144.
Mathematical Reviews (MathSciNet): MR1508688
S. T. Belinschi and A. Nica, On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution, Indiana Univ. Math. J. 57 (2008), 1679--1713.
Mathematical Reviews (MathSciNet): MR2440877
Zentralblatt MATH: 1165.46033
Digital Object Identifier: doi:10.1512/iumj.2008.57.3285
S. T. Belinschi and A. Nica, Free Brownian motion and evolution towards $\boxplus$-infinite divisibility for $k$-tuples, Internat. J. Math. 20 (2009), 309--338.
Mathematical Reviews (MathSciNet): MR2500073
Zentralblatt MATH: 1173.46306
Digital Object Identifier: doi:10.1142/S0129167X09005303
M. Bożejko and W. Bryc, A quadratic regression problem for two-state algebras with application to the central limit theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 231--249.
Mathematical Reviews (MathSciNet): MR2541395
Zentralblatt MATH: 05592929
Digital Object Identifier: doi:10.1142/S0219025709003616
M. Bożejko, M. Leinert and R. Speicher, Convolution and limit theorems for conditionally free random variables, Pacific J. Math. 175 (1996), 357--388.
Mathematical Reviews (MathSciNet): MR1432836
Zentralblatt MATH: 0874.60010
Project Euclid: euclid.pjm/1102353149
M. Bożejko and R. Speicher, $\psi$-independent and symmetrized white noises, Quantum probability & related topics, QP-PQ, vol. VI, World Scientific, River Edge, NJ, 1991, pp. 219--236.
Mathematical Reviews (MathSciNet): MR1149828
T. Cabanal-Duvillard and V. Ionescu, Un théorème central limite pour des variables aléatoires non-commutatives, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 1117--1120.
Mathematical Reviews (MathSciNet): MR1614040
Digital Object Identifier: doi:10.1016/S0764-4442(97)88716-2
D. Damanik, R. Killip and B. Simon, Perturbations of orthogonal polynomials with periodic recursion coefficients, to appear in Ann. Math.; available at arXiv:math /0702388v2 [math.SP], 2009.
U. Franz, Multiplicative monotone convolutions, Quantum probability, Banach Center Publ., vol. 73, Polish Acad. Sci., Warsaw, 2006, pp. 153--166.
Mathematical Reviews (MathSciNet): MR2423123
Zentralblatt MATH: 1109.46054
Digital Object Identifier: doi:10.4064/bc73-0-10
U. Franz, Boolean convolution of probability measures on the unit circle, Analyse et probabilités (J. Faraut, P. Biane and H. Ouerdiane, eds.), Séminaires et Congrès, vol. 16, Soc. Math. France, Paris, 2008, pp. 83--94.
Y. Kato, Mixed periodic Jacobi continued fractions, Nagoya Math. J. 104 (1986), 129--148.
Mathematical Reviews (MathSciNet): MR0868441
Zentralblatt MATH: 0604.10028
Project Euclid: euclid.nmj/1118780556
F. Lehner, Cumulants in noncommutative probability theory. I. Noncommutative exchangeability systems, Math. Z. 248 (2004), 67--100.
Mathematical Reviews (MathSciNet): MR2092722
Zentralblatt MATH: 1089.46040
Digital Object Identifier: doi:10.1007/s00209-004-0653-0
R. Lenczewski, Unification of independence in quantum probability, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 383--405.
Mathematical Reviews (MathSciNet): MR1638097
Zentralblatt MATH: 0928.46044
Digital Object Identifier: doi:10.1142/S021902579800020X
W. Młotkowski, Free probability on algebras with infinitely many states, Probab. Theory Related Fields 115 (1999), 579--596.
Mathematical Reviews (MathSciNet): MR1728922
Zentralblatt MATH: 0941.60020
Digital Object Identifier: doi:10.1007/s004400050250
W. Młotkowski, Operator-valued version of conditionally free product, Studia Math. 153 (2002), 13--30.
Mathematical Reviews (MathSciNet): MR1948925
Zentralblatt MATH: 1036.46044
Digital Object Identifier: doi:10.4064/sm153-1-2
A. Nica and R. Speicher, Lectures on the combinatorics of free probability, London Math. Soc. Lecture Note Ser., vol. 335, Cambridge Univ. Press, Cambridge, 2006.
Mathematical Reviews (MathSciNet): MR2266879
Zentralblatt MATH: 1133.60003
F. Oravecz, The number of pure convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 (2005), 327--355.
Mathematical Reviews (MathSciNet): MR2172303
Zentralblatt MATH: 1091.46043
Digital Object Identifier: doi:10.1142/S0219025705002001
M. Popa, Multilinear function series in conditionally free probability with amalgamation, Commun. Stoch. Anal. 2 (2008), 307--322.
Mathematical Reviews (MathSciNet): MR2446696
L. Verde-Star, Polynomial sequences of interpolatory type, Stud. Appl. Math. 88 (1993), 153--172.
Mathematical Reviews (MathSciNet): MR1204869
Zentralblatt MATH: 0774.41007
D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory. V. Noncommutative Hilbert transforms, Invent. Math. 132 (1998), 189--227.
Mathematical Reviews (MathSciNet): MR1618636
Zentralblatt MATH: 0930.46053
Digital Object Identifier: doi:10.1007/s002220050222
D. V. Voiculescu, K. J. Dykema and A. Nica, Free random variables, A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, CRM Monograph Series, vol. 1, Amer. Math. Soc., Providence, RI, 1992.
Mathematical Reviews (MathSciNet): MR1217253
H. Yoshida, The weight function on non-crossing partitions for the $\Delta$-convolution, Math. Z. 245 (2003), 105--121.
Mathematical Reviews (MathSciNet): MR2023956
Zentralblatt MATH: 1040.46045
Digital Object Identifier: doi:10.1007/s00209-003-0529-8