Fuss-Catalan numbers in noncommutative probabilityth-matrix2014/talks/Mlotkowski_W.pdf · Lambert...

Fuss-Catalan numbers in noncommutativeprobability

Wojtek M lotkowski (Wroc law)

Krakow, 05.07.2014

Wojtek M lotkowski () Fuss-Catalan numbers 05.07.2014 1 / 31

Abstract

The generalized Fuss-Catalan numbers are defined by(np+r

n

)r

np+r , wherep, r are real parameters. They are moments of the Marchenko-Pastur lawMP for p = 2, r = 1 (Catalan numbers) and, more generally, of themultiplicative free power MP�p−1 for r = 1, p > 1. I will presentproperties of these numbers, of their generating functions and ofcorresponding probability distributions and relations with noncommutativeprobability.


1. Combinatorics

Graham, Knuth, Patashnik, Concrete mathematics, Addison-Wesley1994.

The Fuss numbers are defined as(np + 1

n

)1

np + 1, (1)

where p is a real parameter and the generalized binomial coefficient isdefined by (

a

n

):=

a(a− 1) . . . (a− n + 1)

n!. (2)

If p is natural then Fuss numbers have several combinatorialinterpretations:


(1) The number of all such sequences (a1, a2, . . . , anp) that:(a) ai ∈ {1, 1− p}, (b) a1 + a2 + . . .+ as ≥ 0 for all s such that1 ≤ s ≤ np and (c) a1 + a2 + . . .+ anp = 0. Such sequences can berepresented as special Dyck paths on the plane.

(2) The number of such noncrossing partitions π of the set{1, 2, . . . , pn} that every block V ∈ π has p elements.

(3) The number of rooted trees with pn edges, such that everyinternal node has exactly p sons.

The case p = 2, Catalan numbers(2n+1

n

)1

2n+1 , is of particular interest.On the homepage of Richard P. Stanley there are 207 combinatorialinterpretations.

For more general notion, seeDrew Armstrong, Generalized noncrossing partitions and combinatorics ofCoxeter groups, Memoirs of AMS 949 (2009).


2. Generating function

The generating function, studied by Lambert (1750),

Bp(z) :=∞∑n=0

(np + 1

n

)1

np + 1zn, (3)

is convergent in some neighborhood of 0 and and satisfies

Bp(z) = 1 + zBp(z)p. (4)

This means, that Bp(z) is the inverse function to the map w 7→ w−1wp in a

neighborhood of 1, i.e.

Bp(z(1 + z)−p

)= 1 + z (5)

around z = 0.


These functions Bp also satisfy a remarkable composition relation:

Bp(z) = Bp−r(zBp(z)r

). (6)

Another formula:

Bp(z)1+r

(1− p)Bp(z) + p=∞∑n=0

(np + r

n

)zn. (7)


Lambert formula

Lambert (1770) found formula for Taylor expansion of the powers Bp(z)r :

Bp(z)r =∞∑n=0

(np + r

n

)r

np + rzn. (8)

We will call the coefficients(np+r

n

)r

np+r the two-parameter Fussnumbers, or Raney numbers.If p, r are natural then

(np+rn

)r

np+r is the number of all such sequences(a1, a2, . . . , anp+r ) that:(a) ai ∈ {1, 1− p},(b) a1 + a2 + . . .+ as > 0 for all s such that 1 ≤ s ≤ np + r and(c) a1 + a2 + . . .+ anp+r = r .

Graham, Knuth, Patashnik, Concrete mathematics, Addison-Wesley1994.


3. Positive definiteness

A sequence {an}∞n=0 is called positive definite if we have∑i ,j≥0

ai+jcicj ≥ 0 (9)

for an arbitrary sequence {ci}i≥0 of real numbers with finite support (i.e.ci = 0 except for finitely many values of i). This is equivalent, that{an}∞n=0 is a moment sequence for some positive measure µ on R, i.e.

an =

∫R

xn dµ(x). (10)


Positive definiteness of the Fuss numbers

For which parameters p, r ∈ R the sequence(np+r

n

)r

np+r is positivedefinite? If this is the case, the corresponding probability measure will bedenoted µ(p, r).

The Catalan numbers are moments of the Marchenko-Pasturdistribution MP = µ(2, 1):(

2n + 1

n

)1

2n + 1=

∫ 4

0xn 1

2π

√4− x

xdx . (11)

Since (−1)n(np+r

n

)r

np+r =(n(1−p)−r

n

) −rn(1−p)−r , the measure µ(1− p,−r) is

just reflection of µ(p, r). Also the case r = 0 is not interesting becauseµ(p, 0) = δ0. Therefore from now on we assume that p, r > 0.


Theorem

The sequence(mp+r

m

)r

mp+r is positive definite if and only if p ≥ 1,0 < r ≤ p.

The “if” part was first proved by M lotkowski (2010) using freemultiplicative convolution, then in M lotkowski, Penson, Zyczkowski(2013) by using Mellin convolutions of beta measures.The “only if” part was proved by M lotkowski, Penson (2013).Alternative proofs are given by Liu-Pego (2014), Forrester-Liu (2014).The corresponding probability measure µ(p, r) has support[0, pp(p − 1)1−p] and is absolutely continuous.For rational p > 0 the density function Wp,r (x) can be described in termsof the Meijer G -functions (M lotkowski, Penson, Zyczkowski 2013).


M lotkowski, Fuss-Catalan numbers in noncommutative probability,Documenta Mathematica 15 (2010), 939-955.

M lotkowski, Penson, Zyczkowski, Densities of the Raney distributions,Documenta Mathematica 18 (2013), 1573-1596.

M lotkowski, Penson, Probability distributions with binomial moments,Infinite Dimensional Analysis, Quantum Probability and RelatedTopics, 17/2, 2014.

Haagerup, Moller, The law of large numbers for the free multiplicativeconvolution, arXiv:1211.4457 (2012).

Thorsten Neuschel Plancherel-Rotach formulae for averagecharacteristic polynomials of products of Ginibre random matrices andthe Fuss-Catalan distribution, arXiv:1311.0365 (2013).

Peter J. Forrester, Dang-Zheng Liu Raney distributions and randommatrix theory, arXiv:1404.5759 (2014).

Jian-Guo Liu, Robert L. Pego, On generating functions of Hausdorffmoment sequences, arXiv:1401.8052 (2014).


Formula with Meijer G-function

Theorem

For p = k/l > 1, 0 < r ≤ p we have

Wp,r (x) =rpr

x(p − 1)r+1/2√

2kπG k,0k,k

(x l

c(p)l

∣∣∣∣α1, . . . , αk

β1, . . . , βk

), (12)

x ∈ (0, c(p)), where c(p) = pp(p − 1)1−p and the parameters αj , βj aregiven by

αj =

j

lif 1 ≤ j ≤ l ,

r + j − l

k − lif l + 1 ≤ j ≤ k,

(13)

βj =r + j − 1

k, 1 ≤ j ≤ k . (14)


Implicit formula

Theorem

Wp,r (fp(φ)) =sinφ sin rφ (sin(p − 1)φ)p−r−1

π (sin pφ)p−r, (15)

where for 0 < φ < π/p

fp(φ) =(sin pφ)p

sinφ (sin(p − 1)φ)p−1. (16)

Uffe Haagerup, Soren Moller, 2012 (r = 1)

Jian-Guo Liu, Robert L. Pego, 2014 (r = 1).

Thorsten Neuschel, 2013 (r = 1).

Peter J. Forrester, Dang-Zheng Liu, 2014 (general case).


The case p=2

For p = 2, r > 0, the function W2,r is

W2,r (x) =sin(

r · arccos√

x/4)

πx1−r/2, (17)

x ∈ (0, 4). In particular, for r = 1/2, 1, 3/2, 2 we have

W2,1/2(x) =

√2−√

x

2πx3/4, (18)

W2,1(x) =1

2π

√4− x

x, (19)

W2,3/2(x) =

(√x + 1

)√2−√

x

2πx1/4, (20)

W2,2(x) =1

2π

√x(4− x). (21)


The case p=3

Theorem

W3,1(x) =3(

1 +√

1− 4x/27)2/3

− 22/3x1/3

24/331/2πx2/3(

1 +√

1− 4x/27)1/3

, (22)

W3,2(x) =9(

1 +√

1− 4x/27)4/3

− 24/3x2/3

25/333/2πx1/3(

1 +√

1− 4x/27)2/3

(23)

and, finally, W3,3(x) = x ·W3,1(x), with x ∈ (0, 27/4).

Penson, Solomon, Coherent states from combinatorial sequences,Quantum theory and symmetries, Krakow 2001.Penson, Zyczkowski, Product of Ginibre matrices: Fuss-Catalan andRaney distributions Phys. Rev. E 83 (2011).


The case p=3/2,r=1/2

W3/2,1/2(x) =

(1 +

√1− 4x2/27

)2/3−(

1−√

1− 4x2/27)2/3

25/33−1/2πx2/3. (24)

The dilation D2µ(3/2, 1/2), with the density W3/2,1/2(x/2)/2, is known asthe Bures distribution.Sommers, Zyczkowski, Statistical properties of random density matrices,J. Phys. A: Math. Gen. 37 (2004).


The case p=3/2, r=1

MP�1/2 the free multiplicative square root of MP

W3/2,1(x) = 31/2

(1 +

√1− 4x2/27

)1/3−(

1−√

1− 4x2/27)1/3

24/3πx1/3(25)

+31/2x1/3

(1 +

√1− 4x2/27

)2/3−(

1−√

1− 4x2/27)2/3

25/3π


5. Relations of the Fuss numbers with the free, Booleanand monotonic convolution

M lotkowski, Fuss-Catalan numbers in noncommutative probability,Documenta Mathematica 15 (2010).

Classical convolution of probability measures on R:

µ ∗ ν(E ) :=

∫Rµ(E − y) dν(y). (26)

Convolutions coming from noncommutative probability do not have suchdirect description.For a compactly supported measure µ on R we define its momentgenerating function:

Mµ(z) :=

∫R

1

1− xzdµ(x) =

∞∑n=0

zn

∫R

xn dµ(x). (27)


Additive free convolution, Voiculescu 1986

Additive free convolution is defined in the following way. For aprobability measure µ we define its free R-transform Rµ(z) by theformula:

Mµ(z) = Rµ(zMµ(z)) + 1. (28)

If Rµ(z) =∑∞

k=1 rk(µ)zk then rk(µ) are called free cumulants of µ. Thenthe free convolution µ� ν can be defined as the unique probabilitymeasure which satisfies

Rµ�ν(z) = Rµ(z) + Rν(z). (29)

We also define free power µ�t by Rµ�t (z) := tRµ(z). This is well definedat least for t ≥ 1.If µ�t is defined for all t > 0 then we say that µ is infinitely divisiblewith respect to the additive free convolution.


As a consequence of the formula Bp(z) = Bp−r(zBp(z)r

)we have

Theorem

For the free additive transform of µ(p, r) we have

Rµ(p,r)(z) = Bp−r (z)r − 1, (30)

hence the free cumulants of µ(p, r) are equal to(n(p−r)+r

n

)r

n(p−r)+r .

Note that if 0 ≤ 2r ≤ p, r + 1 ≤ p then the cumulants of µ(p, r) aremoments of the measure µ(p − r , r), which implies:

Corollary

If 0 ≤ 2r ≤ p, r + 1 ≤ p then µ(p, r) is �-infinitely divisible.


Free multiplicative convolution, Voiculescu 1986

The free S-transform (or the free multiplicative transform) of aprobability measure µ on R+ := [0,+∞) is defined by the relation

Rµ(zSµ(z)) = z or Mµ

(z(1 + z)−1Sµ(z)

)= 1 + z . (31)

Then the multiplicative free convolution µ1 � µ2 and themultiplicative free power µ�t are defined by

Sµ1�µ2(z) := Sµ1(z)Sµ2(z) and Sµ�t (z) := Sµ(z)t , (32)

the latter is well defined at least for t ≥ 1.From the relation Bp (z(1 + z)−p) = 1 + z we get:


Theorem

For r 6= 0 the S-transform of the measure µ(p, r) is equal to

Sµ(p,r)(z) =(1 + z)

1r − 1

z

(1 + z

) r−pr . (33)

Consequently

µ(1 + p1, 1)� µ(1 + p2, 1) = µ(1 + p1 + p2, 1), (34)

and more generally

µ(p1, r)� µ(1 + p2, 1) = µ(p1 + rp2, r). (35)

We have alsoµ(1 + p, 1)�t = µ(1 + tp, 1). (36)

The Fuss numbers(np+1

n

)1

np+1 are moments of MP�p−1, p ≥ 1.


Boolean convolution, Bozejko, Speicher, Woroudi

The Boolean convolution µ1 ] µ2 and the Boolean power µ]t can bedefined by putting

1

Mµ1]µ2(z)=

1

Mµ1(z)+

1

Mµ2(z)− 1, (37)

Mµ]t (z) =Mµ(z)

(1− t)Mµ(z) + t, (38)

the latter is well defined for all t > 0.From the formula

Bp(z)1+r

(1− p)Bp(z) + p=∞∑n=0

(np + r

n

)zn. (39)

we see that for p ≥ 1 the numbers(npn

)are moments of µ(p, 1)]p.


A by-product: Consider the dilation D1/pµ(p, 1)]p.

Its moments are(npn

)1pn . It is easy to check that(

np

n

)1

pn→ nn

n!(40)

as p →∞. This proves that the sequence nn

n! is positive definite.The corresponding measure was described by Sakuma and Yoshida.

Noriyoshi Sakuma, Hiroaki Yoshida, New limit theorems related to freemultiplicative convolution, Studia Math. 214 (2013), 251-264.


Monotonic convolution, Muraki 2000

The Monotonic convolution is an associative, noncommuting operationµ1 B µ2 = µ on probability measures on R which is defined by

Mµ(z) = Mµ1

(zMµ2(z)

)·Mµ2(z). (41)

Then the formula Bp(z) = Bp−r(zBp(z)r

)leads to:

µ(a, b) B µ(a + r , r) = µ(a + r , b + r), (42)

a ≥ 1, 0 ≤ b ≤ a, r > 0.

Corollary

If 0 ≤ r ≤ p− 1 then the sequence{(mp+r

m

)}∞m=0

is positive definite as themoment sequence of

µ(p − r , 1)]p B µ(p, r).


In fact we have:

Theorem

The sequence{(mp+r

m

)}∞m=0

is positive definite if and only if p ≥ 1 and−1 ≤ r ≤ p − 1.

M lotkowski, Penson, Probability distributions with binomial moments,Infinite Dimensional Analysis, Quantum Probability and Related Topics,17/2, 2014.

Denote the corresponding measure by ν(p, r).We know already that if 0 ≤ r ≤ p − 1 then

ν(p, r) = µ(p − r , 1)]p B µ(p, r).


For c > 0 define probability measure η(c) by

η(c) := c · xc−1 dx , x ∈ [0, 1].

with moments{

cn+c

}∞n=0

.

From the formula(np + r − 1

n

)· r

n(p − 1) + r=

(np + r

n

)r

np + r

we get Mellin convolution relation:

Theorem

For p > 1, 0 < r ≤ p we have

ν(p, r − 1) ◦ η (r/(p − 1)) = µ(p, r).


Theorem

For p > 1 we have

ν(p,−1) = Dc(p)

(1

pδ0 +

p − 1

pδ1

)�p, (43)

where c(p) = pp(p − 1)1−p, and

ν(p, 0) = Dp

((1

pδ0 +

p − 1

pδ1

)�p/(p−1))�p−1

. (44)

D denotes dilation: Dc(µ)(E ) := µ(

1c E).


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