Documenta Math. 939 Fuss-Catalan Numbers in Noncommutative Probability

We prove that if p, r ∈ R, p ≥ 1 and 0 ≤ r ≤ p then the Fuss-Catalan sequence ( mp+r m ) r mp+r is positive definite. We study the family of the corresponding probability measures μ(p, r) on R from the point of view of noncommutative probability. For example, we prove that if 0 ≤ 2r ≤ p and r + 1 ≤ p then μ(p, r) is ⊞-infinitely divisible. As a by-product, we show that the sequence m m m! is positive definite and the corresponding probability measure is ⊠-infinitely divisible. 2010 Mathematics Subject Classification: Primary 46L54; Secondary 44A60, 60C05


Introduction
For natural numbers m, p, r let A m (p, r) denote the number of all sequences (a 1 , a 2 , . . ., a mp+r ) such that: (1) a i ∈ {1, 1 − p}, (2) a 1 + a 2 + . . .+ a s > 0 for all s such that 1 ≤ s ≤ mp + r and (3) a 1 + a 2 + . . .+ a mp+r = r.It turns out that this is given by the two-parameter Fuss-Catalan numbers (2.1) (see [5,13]).Note that the right hand side of (2.1) allows us to define A m (p, r) for all real parameters p and r.In particular, the Catalan numbers A m (2, 1) are known as moments of the Marchenko-Pastur distribution: which in the free probability theory plays the role of the Poisson measure.In this paper we are going to study the question for which parameters p, r ∈ R the sequence {A m (p, r)} ∞ m=0 is positive definite, i.e. is the moment sequence of some probability measure (which we will denote µ(p, r)).Recently T. Banica, S. T. Belinschi, M. Capitaine and B. Collins [1] showed that if p > 1 then {A m (p, 1)} ∞ m=0 is the moment sequence of a probability measure which can be expressed as the multiplicative free power π ⊠p−1 .We are going to prove that if p, r ∈ R, p ≥ 1 and 0 ≤ r ≤ p then {A m (p, r)} ∞ m=0 is the moment sequence of a unique probability measure µ(p, r) which has compact support contained in [0, ∞).Moreover, if 0 ≤ 2r ≤ p and r + 1 ≤ p then µ(p, r) is infinitely divisible with respect to the free convolution ⊞.In some particular cases we are able to determine the multiplicative free convolution, the boolean power and the monotonic convolution of the measures µ(p, r).We will also prove that if 0 ≤ r ≤ p − 1 then the sequence mp+r m ∞ m=0 is positive definite and the corresponding probability measure can be expressed as µ(p−r, 1) ⊎p ⊲ µ(p, r), where ⊎ and ⊲ denote the boolean and the monotonic convolution, respectively.The paper is organized as follows.In Section 2 we prove three combinatorial identities.Then we use them to derive some formulas for the generating functions.In Section 4 we regard the numbers A m (p, r) as moments of a probability quasi-measure µ(p, r) (by this we mean a linear functional µ : R[x] → R satisfying µ(1) = 1).On the class of probability quasi-measures one can introduce the free, boolean and monotonic convolutions in combinatorial way.The class of compactly supported probability measures on R, regarded as a subclass of the former, is closed under these operations.We prove some formulas involving the probability quasi measures µ(p, r), for example we find the free R-and S-transforms (4.8), (4.11), the boolean powers µ(p, 1) ⊎t (4.18) and, in special cases, the multiplicative free (4.12), (4.13), (4.14) and the monotonic convolution (4.20) of the measures µ(p, r).In Section 5 we prove that if p ≥ 1 and 0 ≤ r ≤ p then µ(p, r) is a measure (we conjecture that this condition is also necessary for p, r > 0).The proof involves the multiplicative free convolution ⊠.Moreover, we show that if 0 ≤ 2r ≤ p and r + 1 ≤ p then µ(p, r) is ⊞-infinitely divisible.In the final part we extend our results to the dilations of the measures µ(p, r), with parameter h > 0. Taking the limit with h → 0 we prove in particular that the sequence m m m! ∞ m=0 is positive definite and the corresponding probability measure ν 0 is ⊠-infinitely divisible.

Some combinatorial identities
We will work with the two-parameter Fuss-Catalan numbers (see [5,13]) defined by: A 0 (p, r) := 1 and It is also known (see [13]) that (2.4) Now we are going to prove three identities, valid for c, d, p, r, t ∈ R, which will be needed later on.
Proposition 2.1. (2.5) Proof.It is easy to check that the formula is true for m = 0 and m = 1.
Denoting the left hand side by S m (p, r, c, d) we have from (2.2): so that we have Fix m and assume that (2.5) holds for m − 1.Now we prove that for m it holds for every natural c.Indeed, it holds for c = 0 and if it does for c − 1 then, by assumption and by (2.2): Documenta Mathematica 15 (2010) 939-955 which proves that the statement is true for c.Therefore it holds for all natural c.Now we note that both sides of (2.5) are polynomials on c of order m, therefore the formula holds for all c ∈ R, which completes the inductive step.
(1 − t) Proof.Using first (2.4) and then (2.2) we obtain: Proof.Denoting the left hand side by T m (p, r) we use (2.2) and get Now we proceed as in the proof of (2.5), using the binomial identity

Generating functions
In this part we are going to study the generating functions which are convergent in some neighborhood of 0 (to observe this one can use the inequality and apply the Cauchy's radical test).From (2.4) and (2.3) we have Indeed, denoting the right hand side of (3.2) by A p,r (z) we have A p,1 (z) = B p (z) and, by (2.4),A p,r (z) • A p,s (z) = A p,r+s (z), which implies that A p,r (z) = B p (z) r .Taking r = p and applying (2.3) we get (3.3).Now we are going to interpret formulas (2.5), (2.6), (2.7) in terms of these generating functions.
Proposition 3.1.For any real parameters p, r we have Proof.First we note that if Note that in the proof we applied (2.5)only with c = 1 and d = 0.For p, r, t ∈ R we denote Proposition 3.2.For p, r, t ∈ R we have in particular: Proof.Using (2.6) we can verify that which proves (3.8).Formulas (3.9) and (3.10) are consequences of (2.7) and (3.4).
Proposition 3.3.In some neighborhood of 0 we have and more generally, for r = 0 we have Documenta Mathematica 15 (2010) 939-955 Proof.Since we have B p (0) = 1 and B ′ p (0) = 1, there is a function f p defined on a neighborhood of 0 such that f p (0) = 0 and B(f (3.11) and taking the r-th power we obtain (3.12).
Remark.Note that (3.11) leads to an analytic proof of (3.4).Namely, substituting in (3.4) Finally we note a symmetry possessed by our generating functions.Proposition 3.4.For p, r, t ∈ R we have

Relations with noncommutative probability
By a probability quasi-measure we will mean a linear functional µ on the set R[x] of polynomials with real coefficients, satisfying µ(1) = 1.In the case when µ is given by µ(P ) = P (t) d µ(t) for some probability measure µ on R we will identify µ with µ and say that µ is proper or is just a probability measure.A probability quasi-measure µ is uniquely determined by its moment sequence {µ(x m )} ∞ m=0 .It is proper if and only if its moment sequence is positive definite, i.e. if ∞ i,j=0 µ(x i+j )α i α j ≥ 0 holds for every sequence {α i } ∞ i=0 of real numbers, with only finitely many nonzero entries.All probability measures encountered in this paper are compactly supported and therefore uniquely determined by their moment sequences.For a probability quasi-measure µ we define its moment generating function, which is the (at least formal) power series and its reflection µ by µ(x m ) := (−1) m µ(x m ) or, equivalently, M µ (z) := M µ (−z).If µ is a probability measure then so is µ and then we have µ(X) = µ(−X) for every Borel subset of R.
First we note that Proposition 3.4 leads to Proposition 4.1.
There are several convolutions of probability quasi-measures, apart from the classical one: ), which are related to various notions of independence (namely, the free, boolean and the monotonic independence) in noncommutative probability.1. Free convolution (see [2,15,11]) is defined in the following way.For a probability quasi-measure µ we define its free R-transform (or the additive free transform) R µ (z) by the formula: The free cumulants r m (µ) are defined as the coefficients of the Taylor expansion R µ (z) = ∞ k=1 r k (µ)z k (combinatorial relation between moments and free cumulants is described in [11] and [4]).Then the free convolution µ ⊞ ν can be defined as the unique probability quasi-measure which satisfies We also define free power µ ⊞t , t > 0, by R µ ⊞t (z) := tR µ (z).As a consequence of (4.6) and (3.4) we obtain: Proposition 4.2.For the free additive transform of µ(p, r) we have The free S-transform (or the free multiplicative transform) of a quasi-measure µ, with µ(x 1 ) = 0, is defined by the relation Documenta Mathematica 15 (2010) 939-955 Then the multiplicative free convolution µ 1 ⊠ µ 2 and the multiplicative free power µ ⊠t , t > 0, are defined by (4.10) S µ1⊠µ2 (z) := S µ1 (z)S µ2 (z) and S µ ⊠t (z) := S µ (z) t .
3. Monotonic convolution (see [10]) is an associative, noncommuting operation ⊲ which is defined by:  In the next section we are going to study which of the probability quasimeasures µ(p, r) and ν(p, r, t) are actually probability measures.For this purpose we will use some of the the following facts, which are available in literature (see [15,11,14,10,6,7]): The class of all compactly supported probability measures on R is closed under the free, boolean, and monotonic convolution and also under taking the powers µ ⊞s , µ ⊎t , for s ≥ 1, t > 0.Moreover, the class of probability measures with compact support contained in [0, ∞) is closed under the free, multiplicative free, boolean and monotonic convolution and also under taking the powers µ ⊞s , µ ⊠s and µ ⊎t for s ≥ 1 and t > 0. A probability measure µ on R (resp.on [0, ∞)) is called ⊞-infinitely divisible (resp.⊠-infinitely divisible) if µ ⊞t (resp.µ ⊠t ) is a probability measure for every t > 0. If µ has compact support and r m (µ) are its free cumulants then µ is ⊞-infinitely divisible if and only if the sequence {r m+2 (µ)} ∞ m=0 is positive definite.

Positivity
The aim of this section is to study which of the quasi measures µ(p, r) and ν(p, r, t) are actually measures, i.e. for which parameters p, r, t ∈ R the corresponding sequence is positive definite.We start with Proof.We know already that π = µ(2, 1) is the free Poisson law (1.1).Then, as was noted in [1], π is ⊠-infinitely divisible and for s > 0 we have π ⊠s = µ(1 + s, 1).Hence if p ≥ 1 then µ(p, 1) is a probability measure with a compact support contained in [0, ∞).By (2.3) it implies that the sequence A m (p, p) = A m+1 (p, 1) is also positive definite, namely we have for any continuous function f on R. Hence µ(p, p), p ≥ 1, is a probability measure with a compact support contained in [0, ∞).For 1 ≤ r ≤ p we apply (4.13) to obtain: µ(p, r) = µ(r, r) ⊠ µ(p/r, 1), which proves the first statement for the sector 1 ≤ r ≤ p.For r ∈ (0, 1) the measure µ(1, r) is related to the Euler beta function which means that µ(1, r) = µ r .Now for s ≥ 0 we have which proves the first statement for (p, r) ∈ [1, +∞) × (0, 1).It remains to note that µ(p, 0) = δ 0 for every p ∈ R.
The second statement is a consequence of (4.4).
We conjecture that the last theorem fully characterizes the set of parameters p, r ∈ R for which µ(p, r) is a measure (apart from the trivial case µ(p, 0) = δ 0 ).It is easy to check that A 0 (p, r)A 2 (p, r) − A 1 (p, r) 2 = r(2p − 1 − r)/2, hence a necessary condition for positive definiteness of the sequence Remark.According to Penson and Solomon [12]: More generally, for µ(p, 1) with p ∈ N we refer to [8].
Proof.By Theorem 13.16 in [11], a compactly supported probability measure µ, with free cumulants r m (µ), is ⊞-infinitely divisible if and only if the sequence {r m+2 (µ)} ∞ m=0 is positive definite.Then it is sufficient to refer to (4.8) and to note that the numbers A m+2 (p − r, r) are the moments of the measure x 2 dµ(p − r, r)(x).
A measure ν on R is called symmetric if ν = ν.For a probability quasi-measure µ define its symmetrization µ s by M µ s (z) := M µ z 2 .If µ is a probability measure with support contained in [0, ∞) then µ s is a symmetric measure on R, which satisfies R f (t 2 ) dµ s (t) = R f (t) dµ(t) for every compactly supported continuous function on R. Denote by µ s (p, r) and ν s (p, r, t) the symmetrization of µ(p, r) and ν(p, r, t).Then, by (3.4) and (4.9), for the free additive transform we have In the same way as Corollary 5.2 one can prove Let us record some formulas: Here we illustrate the main results concerning the measures µ(p, r).

Dilations
For a probability quasi-measure µ we define its dilation with parameter c > 0 by (D c µ)(x m ) := c m µ(x m ).Then for the moment generating function we have: M Dcµ (z) = M µ (cz) and similarly for the free R-transform: R Dcµ (z) = R µ (cz), while for the S-transform we have S Dcµ (z) = 1 c S µ (z).If µ is proper then we have (D c µ)(X) = µ 1 c X for every Borel subset X of R. In this part we are going to study dilations of the measures µ(p, r) and ν(p, r, t) and their limits as the parameter goes to 0. For h ≥ 0 and a, p, r ∈ R define sequences (a − ih), (6.1) (mp + r − ih), (6.2) with A 0 (p, r, h) := 1.In particular A m (p, r, h) = r mp+r mp+r m h whenever mp + r = 0.Then, for h ≥ 0 and p, r, t ∈ R define probability quasi-measures: and their moment generating functions B p,r,h (z) and D p,r,t,h (z) respectively.Note that if h > 0 then A m (p, r, h) = h m A m (p/h, r/h) and hence these probability quasi measures can be represented as dilations: µ(p, r, h) = D h µ(p/h, r/h), (6.5) ν(p, r, t, h) = D h ν(p/h, r/h, t/h).(6.6) Therefore the corresponding moment generating functions are B p,r,h (z) = B p/h (hz) r/h , (6.7) D p,r,t,h (z) = D p/h,r/h,t/h (hz) = hB p,h+r,h (z) (h − t)B p,h,h (z) + t .(6.8) These formulas allow us to derive properties of the probability quasi-measures µ(p, r, h) and ν(p, r, t, h) directly from our previous results when h > 0, and, after taking the limit with h → 0, for h = 0. Proposition 6.1.For h > 0 and p, r, t ∈ R B p,h,h (z) = 1 + zhB p,p,h (z), (6.9) log (B p,1,0 (z)) = zB p,p,0 (z), (6.10) D p,r,t,0 (z) = B p,r,0 (z) 1 − ztB p,p,0 (z) .(6.11) Proof.First formula is a consequence of (3.3) and (6.7).Then we have Taking the limit with h → 0 we obtain (6.10).For (6.11) we write use (6.8) and (6.9) to get and then we take limit with h → 0. Proof.For h > 0 the formula is a consequence of (6.6).Then we take limit with h → 0.