
Limit theorems for monotonic convolution and the Chernoff product formula (with
John D. Williams), to be published by International Mathematics Research Notices. A version with more general results is at
arXiv:1209.4260 [math.OA].
Bercovici and Pata showed that the correspondence between classically, freely, and Boolean infinitely divisible distributions holds on the level of limit theorems. We extend this correspondence also to distributions infinitely divisible with respect to the additive monotone convolution. Because of noncommutativity of this convolution, we use a new technique based on the Chernoff product formula. We also study this correspondence for multiplicative monotone convolution, where the BercoviciPata bijection no longer holds.

Free evolution on algebras with two states II, arXiv:1204.0289 [math.OA].
Denote by J the operator of coefficient stripping. We show that for any free convolution semigroup {ν_{t} : t ≥ 0} with finite variance, applying a single stripping produces semicircular evolution with nonzero initial condition, J[ν_{t}] = ρ + σ_{β, γ}^{+ t}, where σ_{β, γ} is the semicircular distribution with mean β and variance γ. For more general freely infinitely divisible distributions τ, expressions of the form ρ + τ^{+ t} arise from stripping μ_{t}, where {(μ_{t}, ν_{t}) : t ≥ 0} form a semigroup under the operation of twostate free convolution. The converse to this statement holds in the algebraic setting. Numerous examples illustrating these constructions are computed. Additional results include the formula for generators of such semigroups.

A characterization of ultraspherical polynomials, arXiv:1108.0914 [math.CA].
This is an expository article, with all details included.
We show that the only orthogonal polynomials with a generating function of the form F(x z  α z^{2}) are the ultraspherical, Hermite, and Chebyshev polynomials of the first kind. The generating function for the Chebyshev case is nonstandard, although it is easily derived from the usual one.

Convolution powers in the operatorvalued framework (with Serban T. Belinschi, Maxime Fevrier, and Alexandru Nica), Trans. Amer. Math. Soc. 365 (2013), 20632097.
We consider the framework of an operatorvalued noncommutative probability space over a unital C^{*}algebra B. We show how for a Bvalued distribution μ one can define convolution powers μ^{+η} (with respect to free additive convolution) and μ^{∪η} (with respect to Boolean convolution), where the exponent η is a suitably chosen linear map from B to B, instead of being a nonnegative real number. More precisely, μ^{∪η} is always defined when η is completely positive, while μ^{+η} is always defined when η  1 is completely positive (with "1" denoting the identity map on B).
In connection to these convolution powers we define an evolution semigroup {B_{η}  η : B → B, completely positive}, related to the Boolean BercoviciPata bijection. We prove several properties of this semigroup, including its connection to the Bvalued free Brownian motion.
We also obtain two results on the operatorvalued analytic function theory related to convolution powers μ^{+η}. One of the results concerns the analytic subordination of the CauchyStieltjes transform of μ^{+η} with respect to the CauchyStieltjes transform of μ. The other one gives a Bvalued version of the inviscid Burgers equation, which is satisfied by the CauchyStieltjes transform of a Bvalued free Brownian motion.

Generators of some noncommutative stochastic processes,
Probab. Theory Related Fields 157 (2013), 777815.
A fundamental result of Biane (1998) states that a process with freely independent increments has the Markov property, but for a process with stationary increments the transition operators are not necessarily stationary. So instead of forming a semigroup with a generator, they form a twoparameter family with a timedependent family of generators. We compute an explicit formula for these generators, in terms of singular integral operators, and prove that the formula holds on a fairly large domain. We also compute the generators for the qBrownian motion, and for the twostate free Brownian motions.

Quantum free YangMills on the plane (with Ambar N. Sengupta), J. Geom. Phys. 62 (2012), 330343.
We construct a freeprobability quantum YangMills theory on the two dimensional plane, determine the Wilson loop expectation values, and show that
this theory is the N = ∞ limit of U(N) quantum YangMills theory on the plane.

Semigroups of distributions with linear Jacobi parameters (with Wojciech Młotkowski), J. Theoret. Probab. 25 (2012), 11731206.
We show that a convolution semigroup {μ_{t}} of measures has Jacobi parameters polynomial in the convolution parameter t if and only if the measures come from the Meixner class. Moreover, we prove the parallel result, in a more explicit way, for the free convolution and the free Meixner class. We then construct the class of measures satisfying the same property for the twostate free convolution. This class of twostate free convolution semigroups has not been considered explicitly before. We show that it also has Meixnertype properties. Specifically, it contains the analogs of the normal, Poisson, and binomial distributions, has a LahaLukacstype characterization, and is related to the q=0 case of quadratic harnesses.

Twostate free Brownian motions,
J. Funct. Anal. 260 (2011), 541565.
In a twostate free probability space (A, φ, ψ), we define an algebraic twostate free Brownian motion to be a process with twostate freely independent increments whose twostate free cumulant generating function R^{φ, ψ}(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states φ, ψ are normal, and φ is faithful, then there is only a oneparameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to φ = ψ), these processes only exist for finite time.

BochnerPearsontype characterization of the free Meixner class,
Adv. in Appl. Math. 46 (2011), 2545 (special issue in honor of Dennis Stanton).
The operator L_{μ} : f → ∫ (f(x)  f(y))/(x  y) dμ(y) is, for a compactly supported measure μ with an L^{3} density, a closed, densely defined operator on L^{2}(μ). We show that the operator Q = p L_{μ}^{2}  q L_{μ} has polynomial eigenfunctions if and only if μ is a free Meixner distribution. The only time Q has orthogonal polynomial eigenfunctions is if μ is a semicircular distribution. More generally, the only time the operator p (L_{ν} L_{μ})  q L_{μ} has orthogonal polynomial eigenfunctions is when μ and ν are related by a Jacobi shift.

Free infinite divisibility for qGaussians (with Serban Teodor Belinschi, Marek Bożejko, and Franz Lehner), Math. Res. Lett. 17 (2010), 905916.
We prove that the qGaussian distribution introduced by Bożejko and Speicher is freely infinitely divisible for all q between zero and one.

Producttype noncommutative polynomial states, Noncommutative Harmonic Analysis with Applications to Probability II, Banach Center Publ., vol. 89, Polish Acad. Sci. Inst. Math., Warsaw, 2010, pp. 4559.
In [Ans08],
[Ans09], we investigated monic multivariate noncommutative orthogonal polynomials, their recursions, states of orthogonality, and corresponding continued fraction expansions. In this note, we collect a number of examples, demonstrating what these general results look like for the most important states on noncommutative polynomials, namely for various product states. In particular, we introduce a notion of a producttype state on polynomials, which covers all the noncommutative universal products and excludes some other familiar noncommutative products, and which guarantees a number of nice properties for the corresponding polynomials.

Free evolution on algebras with two states,
J. Reine Angew. Math. 638 (2010), 75101.
The key result in the paper concerns two transformations, Φ(ρ, ψ) and B_{t}(ψ) on states on the algebra of noncommutative polynomials, or equivalently on joint distributions of dtuples of noncommuting operators. These transformations are related to free probability: we show that Φ intertwines the action of B_{t} and the free convolution with the semigroup {ρ_{t}}. The maps {B_{t}} were introduced by Belinschi and Nica as a semigroup of transformations such that B_{1} is the bijection between infinitely divisible distributions in the Boolean and free probability theories. They proved the intertwining property above for a singlevariable version of the map Φ and the particular case of the free heat semigroup. The more general twovariable map Φ comes, not from free probability, but from the theory of twostate algebras, also called the conditionally free probability theory, introduced by Bożejko, Leinert, and Speicher. Orthogonality of the cfree versions of the Appell polynomials, investigated in [Ans09], is closely related to the singlevariable map Φ. On the other hand, more general free Meixner families behave well under all the transformations above, and provide clues to their general behavior. Besides the evolution equation, other results include the positivity of the map Φ and descriptions of its fixed points and range.

Appell polynomials and their relatives III. Conditionally free theory,
Illinois J. Math. 53 (2009), 3966.
We extend to the multivariate noncommutative context the descriptions of a "oncestripped" 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 cfree probability theory, which is a version of free probability for algebras with two states, introduced by Bożejko, Leinert, and Speicher. This theory includes as two extreme cases the free and Boolean probability theories. 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 cfree version of the KailathSegall polynomials, their combinatorics, and Hilbert space representations.

Appell polynomials and their relatives II. Boolean theory,
Indiana Univ. Math. J. 58 (2009), 929968.
The Appelltype polynomial family corresponding to the simplest noncommutative derivative operator turns out to be connected with the Boolean probability theory, the simplest of the three universal noncommutative probability theories (the other two being free and tensor/classical probability). The basic properties of the Boolean Appell polynomials are described. In particular, their generating function turns out to have a resolventtype form, just like the generating function for the free Sheffer polynomials. It follows that the Meixner (that is, Sheffer plus orthogonal) polynomial classes, in the Boolean and free theory, coincide. This is true even in the multivariate case. A number of applications of this fact are described, to the BelinschiNica and BercoviciPata maps, conditional freeness, and the LahaLukacs type characterization.
A number of properties which hold for the Meixner class in the free and classical cases turn out to hold in general in the Boolean theory. Examples include the behavior of the Jacobi parameters under convolution, the relationship between the Jacobi parameters and cumulants, and an operator model for cumulants. Along the way, we obtain a multivariate version of the Stieltjes continued fraction expansion for the moment generating function of an arbitrary state with monic orthogonal polynomials.

Monic noncommutative orthogonal polynomials,
Proc. Amer. Math. Soc. 136 (2008), 23952405.
Among all states on the algebra of noncommutative polynomials, we characterize the ones that have monic orthogonal polynomials. The characterizations involve recursion relations, Hankeltype determinants, and a representation as a joint distribution of operators on a Fock space.

Orthogonal polynomials with a resolventtype generating function, Trans. Amer. Math. Soc. 360 (2008), 41254143.
The subject of this paper are polynomials in multiple noncommuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favardtype recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in noncommuting variables with a resolventtype 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 secondorder 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 nontracial infinitely divisible states with orthogonal free Sheffer polynomials.

Free Meixner states,
Commun. Math. Phys. 276 (2007), 863899.
Free Meixner states are a class of functionals on noncommutative polynomials introduced in
[Ans08]. They are characterized by a resolventtype form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a secondorder noncommutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Many of these examples are shown to be trivial, in the sense of being free products of functionals which depend on only a single variable, or rotations of such free products. On the other hand, the multinomial distribution is a free Meixner state and is not a product. Neither is a large class of tracial free Meixner states which are analogous to the simple quadratic exponential families in statistics.

Zimmermann type cancellation in the free Faà di Bruno algebra (with Edward G. Effros and Mihai Popa), J. Funct. Anal. 237 (2006), 76104.
Haiman and Schmitt showed that one can use the antipode S_F of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode S_H of an algebraically free analogue H of F may be used to invert noncommutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for S_F are labeled by trees. By contrast, the noncommuting monomials of S_H are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the ConnesKreimer Hopf algebra.

Linearization coefficients for orthogonal polynomials using stochastic processes,
Ann. Probab. 33 (2005), 114136.
Given a basis for a polynomial ring, the coefficients in the expansion of a product of some of its elements in terms of this basis are called linearization coefficients. These coefficients have combinatorial significance for many classical families of orthogonal polynomials. Starting with a stochastic process and using the stochastic measures machinery introduced by Rota and Wallstrom, we calculate and give an interpretation of linearization coefficients for a number of polynomial families. The processes involved may have independent, freely independent, or qindependent increments. The use of noncommutative stochastic processes extends the range of applications significantly, allowing us to treat Hermite, Charlier, Chebyshev, free Charlier, and Rogers and continuous big qHermite polynomials.

qLévy processes,
J. Reine Angew. Math. 576 (2004), 181207.
We continue the investigation of the Lévy processes on a qdeformed full Fock space started in [Ans01]. First, we show that the vacuum vector is cyclic and separating for the algebra generated by such a process. Next, we describe a chaotic representation property in terms of multiple integrals with respect to diagonal measures, in the style of Nualart and Schoutens. We define stochastic integration with respect to these processes, and calculate their combinatorial stochastic measures. Finally, we show that they generate infinite von Neumann algebras.

Appell polynomials and their relatives,
Int. Math. Res. Not. 2004 n. 65, 34693531.
Maple worksheet used in the Appendix. A more complete version of the paper is at
arXiv:math/0311043 [math.CO].
This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the Appell polynomials when polynomials in noncommuting variables are considered. They also fit well into the framework of free probability. For the free Appell polynomials, a number of combinatorial and "diagram" formulas are proven, such as the formulas for their linearization coefficients. An explicit formula for their generating function is obtained. These polynomials are also martingales for free Lévy processes. For more general free Sheffer families, all the pseudoorthogonal families are described. Another family investigated are the KailathSegall polynomials. These are multivariate polynomials, which share with the Appell polynomials nice combinatorial properties, but are always orthogonal. Their origins lie in the Fock space representations, or in the theory of multiple stochastic integrals. Diagram formulas are proven for these polynomials as well, even in the qdeformed case.

Free martingale polynomials,
J. Funct. Anal. 201 (2003), 228261.
In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the polynomials whose derivatives are martingale polynomials. Finally, we indicate how Rota's finite operator calculus can be modified for the free context.

Itô formula for free stochastic integrals,
J. Funct. Anal. 188 (2002), 292315.
The objects under investigation are the stochastic integrals with respect to free Lévy processes. We define such integrals for squareintegrable integrands, as well as for a certain general class of bounded integrands. Using the product form of the Itô formula, we prove the full functional Itô formula in this context.

Free stochastic measures via noncrossing partitions II,
Pacific J. Math. 207 (2002), 1330.
We show that for stochastic measures with freely independent increments, the partitiondependent stochastic measures of
[Ans00] can be expressed purely in terms of the higher stochastic measures and the higher diagonal measures of the original.

Partitiondependent stochastic measures and qdeformed cumulants, Doc. Math. 6 (2001), 343384.
On a qdeformed Fock space, we define multiple qLévy processes. Using the partitiondependent stochastic measures derived from such processes, we define partitiondependent cumulants for their joint distributions, and express these in terms of the cumulant functional using the number of restricted crossings of P. Biane. In the single variable case, this allows us to define a qconvolution for a large class of probability measures. We make some comments on the Itô table in this context, and investigate the qBrownian motion and the qPoisson process in more detail.

Free stochastic measures via noncrossing partitions,
Adv. Math. 155 (2000), 154179.
We consider free multiple stochastic measures in the combinatorial framework of the lattice of all diagonals of an ndimensional space. In this free case, one can restrict the analysis to only the noncrossing diagonals. We give definitions of what free multiple stochastic measures are, and calculate them for the free Poisson and free compound Poisson processes. We also derive general combinatorial Itôtype relationships between free stochastic measures of different orders. These allow us to calculate, for example, free PoissonCharlier polynomials, which are the orthogonal polynomials with respect to the free Poisson measure.

The linearization of the central limit operator in free probability theory,
Probab. Theory Related Fields 115 (1999), 401416.
We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory, we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear.