Fuss-Catalan numbers in noncommutative probability

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1 Fuss-Catalan numbers in noncommutative probability Wojciech M lotkowski (Wroc law) Bielefeld, Wojciech M lotkowski Fuss-Catalan numbers / 35

2 Abstract The generalized Fuss-Catalan numbers are defined by ( ) np + r r n np + r, where p, r are real parameters. For p = 2, r = 1 (Catalan numbers) they are moments of the Marchenko-Pastur law MP, i.e. ( 2n + 1 n ) 1 2n + 1 = 1 2π 4 0 x n 4 x x dx. More generally, for r = 1, p > 1 they are moments of the multiplicative free power MP p 1. I will present properties of these numbers, of their generating functions and of corresponding probability distributions and relations with noncommutative probability. Wojciech M lotkowski Fuss-Catalan numbers / 35

3 1. Combinatorics Graham, Knuth, Patashnik, Concrete mathematics, Addison-Wesley The Fuss numbers are defined as ( ) np n np + 1, (1) where p is a real parameter and the generalized binomial coefficient is defined by ( ) a a(a 1)... (a n + 1) :=. (2) n n! If p is natural then Fuss numbers have several combinatorial interpretations: Wojciech M lotkowski Fuss-Catalan numbers / 35

4 (1) The number of all such sequences (a 1, a 2,..., a np ) that: (a) a i {1, 1 p}, (b) a 1 + a a s 0 for all s such that 1 s np and (c) a 1 + a a np = 0. Such sequences can be represented 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 every internal node has exactly p sons. ) 1 The case p = 2, Catalan numbers ( 2n+1 n 2n+1, is of particular interest, see the monograph: Richard P. Stanley, Catalan Numbers, Cambridge University Press 2015, contains over 200 combinatorial interpretations of the Catalan numbers Wojciech M lotkowski Fuss-Catalan numbers / 35

5 2. Generating function The generating function, studied by Lambert (1750), B p (z) := ( ) np n np + 1 zn, (3) n=0 is convergent in some neighborhood of 0 and and satisfies B p (z) = 1 + zb p (z) p. (4) This means, that B p (z) is the inverse function to the map w w 1 w p neighborhood of 1, i.e. in a B p ( z(1 + z) p ) = 1 + z (5) around z = 0. Wojciech M lotkowski Fuss-Catalan numbers / 35

6 These functions B p also satisfy a remarkable composition relation: B p (z) = B p r ( zbp (z) r ). (6) Another formula: B p (z) 1+r (1 p)b p (z) + p = n=0 ( np + r n ) z n. (7) Lambert formula (1770) B p (z) r = ( ) np + r r n np + r zn. (8) n=0 Donald Knuth s 20th Annual Christmas Tree Lecture: (3/2)-ary Trees, youtube. Wojciech M lotkowski Fuss-Catalan numbers / 35

7 ) r np+r We will call the coefficients ( np+r n the two-parameter Fuss numbers, or Raney numbers. If p, r are natural then ( ) np+r r n np+r is the number of all such sequences (a 0, a 1, a 2,..., a np+r ) that: (a) a i {1, 1 p}, (b) a 0 + a a s > 0 for all s such that 1 s np + r and (c) a 0 + a a np+r = r. Graham, Knuth, Patashnik, Concrete mathematics, Addison-Wesley Wojciech M lotkowski Fuss-Catalan numbers / 35

8 3. Positive definiteness A sequence {a n } n=0 is called positive definite if we have a i+j c i c j 0 (9) i,j 0 for an arbitrary sequence {c i } i 0 of real numbers with finite support (i.e. c i = 0 except for finitely many values of i). This is equivalent, that {a n } n=0 is a moment sequence for some positive measure µ on R, i.e. a n = x n dµ(x). (10) R Wojciech M lotkowski Fuss-Catalan numbers / 35

9 Positive definiteness of the Fuss numbers For which parameters p, r R the sequence ( np+r n np+r is positive definite? If this is the case, the corresponding probability measure will be denoted µ(p, r). The Catalan numbers are moments of the Marchenko-Pastur distribution MP = µ(2, 1): ( ) 2n n 2n + 1 = x n 1 4 x dx. (11) 0 2π x ) Since ( 1) n( ) np+r r n np+r = ( n(1 p) r n r n(1 p) r, the measure µ(1 p, r) is just reflection of µ(p, r). Also the case r = 0 is not interesting because µ(p, 0) = δ 0. ) r Wojciech M lotkowski Fuss-Catalan numbers / 35

10 From now on we assume that p, r > 0. Theorem The sequence ( ) mp+r m 0 < r p. r mp+r is positive definite if and only if p 1, Special case: (p, r) = (3/2, 1/2)- the dilated Bures distribution. Wojciech M lotkowski Fuss-Catalan numbers / 35

11 The if part was first proved by M lotkowski (2010) using free multiplicative convolution, then in M lotkowski, Penson, Życzkowski (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, p p (p 1) 1 p ] and is absolutely continuous. For rational p > 0 the density function W p,r (x) can be described in terms of the Meijer G-functions (M lotkowski, Penson, Życzkowski 2013). Wojciech M lotkowski Fuss-Catalan numbers / 35

12 M lotkowski, Fuss-Catalan numbers in noncommutative probability, Documenta Mathematica 15 (2010), Liu, Song, Wang, On explicit probability densities associated with Fuss-Catalan numbers, Proc. AMS M lotkowski, Penson, Życzkowski, Densities of the Raney distributions, Documenta Mathematica 18 (2013). M lotkowski, Penson, Probability distributions with binomial moments, Inf. Dim. Analysis, Quantum Prob. and Related Topics, 17/2, Haagerup, Möller, The law of large numbers for the free multiplicative convolution, arxiv: (2012). Thorsten Neuschel Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution, Random Matrices: Theory and Applications, Vol. 03, No. 01, (2014). Peter J. Forrester, Dang-Zheng Liu Raney distributions and random matrix theory, Journal of Statistical Physics, Issue 5. Jian-Guo Liu, Robert L. Pego, On generating functions of Hausdorff moment sequences, arxiv: (2014). Wojciech M lotkowski Fuss-Catalan numbers / 35

13 Formula with Meijer G-function Theorem For p = k/l > 1, 0 < r p we have W p,r (x) = rp r ( x(p 1) r+1/2 2kπ G k,0 x l k,k c(p) l α 1,..., α k β 1,..., β k ), (12) x (0, c(p)), where c(p) = p p (p 1) 1 p and the parameters α j, β j are given by j if 1 j l, α j = l (13) r + j l if l + 1 j k, k l β j = r + j 1, 1 j k. (14) k Wojciech M lotkowski Fuss-Catalan numbers / 35

14 Implicit formula Theorem W p,r (f p (φ)) = where for 0 < φ < π/p f p (φ) = sin φ sin rφ (sin(p 1)φ)p r 1 π (sin pφ) p r, (15) (sin pφ) p. (16) p 1 sin φ (sin(p 1)φ) Uffe Haagerup, Sören Möller, 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). Wojciech M lotkowski Fuss-Catalan numbers / 35

15 The case p=2 For p = 2, r > 0, the function W 2,r is W 2,r (x) = ( sin r arccos ) x/4 πx 1 r/2, (17) x (0, 4). In particular, for r = 1/2, 1, 3/2, 2 we have 2 x W 2,1/2 (x) =, (18) 2πx 3/4 W 2,1 (x) = 1 4 x, (19) 2π x ( ) x x W 2,3/2 (x) = 2πx 1/4, (20) W 2,2 (x) = 1 x(4 x). (21) 2π Wojciech M lotkowski Fuss-Catalan numbers / 35

16 The case p=3 Theorem ( /3 1 4x/27) 2 2/3 x 1/3 W 3,1 (x) = 2 4/3 3 1/2 πx (1 2/3 + ) 1/3, (22) 1 4x/27 ( /3 1 4x/27) 2 4/3 x 2/3 W 3,2 (x) = 2 5/3 3 3/2 πx (1 1/3 + ) 2/3 (23) 1 4x/27 and, finally, W 3,3 (x) = x W 3,1 (x), with x (0, 27/4). Penson, Solomon, Coherent states from combinatorial sequences, Quantum theory and symmetries, Kraków Penson, Życzkowski, Product of Ginibre matrices: Fuss-Catalan and Raney distributions Phys. Rev. E 83 (2011). Wojciech M lotkowski Fuss-Catalan numbers / 35

17 The case p=3/2,r=1/2 W 3/2,1/2 (x) = ( 1 + ) 2/3 ( 1 4x 2 /27 1 ) 2/3 1 4x 2 /27 2 5/3 3 1/2 πx 2/3. (24) The dilation D 2 µ(3/2, 1/2), with the density W 3/2,1/2 (x/2)/2, is known as the Bures distribution. Sommers, Życzkowski, Statistical properties of random density matrices, J. Phys. A: Math. Gen. 37 (2004). Wojciech M lotkowski Fuss-Catalan numbers / 35

18 The case p=3/2, r=1 MP 1/2 the free multiplicative square root of MP W 3/2,1 (x) = 3 1/2 ( 1 + ) 1/3 ( 1 4x 2 /27 1 ) 1/3 1 4x 2 /27 2 4/3 πx 1/3 (25) ( 1 + ) 2/3 ( 1 4x 2 /27 1 ) 2/3 1 4x 2 / /2 x 1/3 2 5/3 π Wojciech M lotkowski Fuss-Catalan numbers / 35

19 Wojciech M lotkowski Fuss-Catalan numbers / 35

22 5. Relations of the Fuss numbers with the free, Boolean and monotonic convolution Classical convolution of probability measures on R: µ ν(e) := µ(e y) dν(y). (26) R Convolutions coming from noncommutative probability do not have such direct description. For a compactly supported measure µ on R we define its moment generating function: M µ (z) := R 1 1 xz dµ(x) = z n n=0 R x n dµ(x). (27) Wojciech M lotkowski Fuss-Catalan numbers / 35

23 Additive free convolution, Voiculescu 1986 Additive free convolution is defined in the following way. For a probability measure µ we define its free R-transform R µ (z) by the formula: M µ (z) = R µ (zm µ (z)) + 1. (28) If R µ (z) = k=1 r k(µ)z k then r k (µ) are called free cumulants of µ. Then the free convolution µ ν can be defined as the unique probability measure which satisfies R µ ν (z) = R µ (z) + R ν (z). (29) We also define free power µ t by R µ t (z) := tr µ (z). This is well defined at least for t 1. If µ t is defined for all t > 0 then we say that µ is infinitely divisible with respect to the additive free convolution. Wojciech M lotkowski Fuss-Catalan numbers / 35

24 As a consequence of the formula B p (z) = B p r ( zbp (z) r ) we have Theorem For the free additive transform of µ(p, r) we have R µ(p,r) (z) = B p 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) are moments of the measure µ(p r, r), which implies: Corollary If 0 2r p, r + 1 p then µ(p, r) is -infinitely divisible. Wojciech M lotkowski Fuss-Catalan numbers / 35

25 Free multiplicative convolution, Voiculescu 1986 The free S-transform (or the free multiplicative transform) of a probability measure µ on R + := [0, + ) is defined by the relation R µ (zs µ (z)) = z or M µ ( z(1 + z) 1 S µ (z) ) = 1 + z. (31) Then the multiplicative free convolution µ 1 µ 2 and the multiplicative free power µ t are defined by S µ1 µ 2 (z) := S µ1 (z)s µ2 (z) and S µ t (z) := S µ (z) t, (32) the latter is well defined at least for t 1. From the relation B p (z(1 + z) p ) = 1 + z we get: Wojciech M lotkowski Fuss-Catalan numbers / 35

26 Theorem For r 0 the S-transform of the measure µ(p, r) is equal to Consequently S µ(p,r) (z) = (1 + z) 1 r 1( ) r p 1 + z r. (33) z and more generally µ(1 + p 1, 1) µ(1 + p 2, 1) = µ(1 + p 1 + p 2, 1), (34) µ(p 1, r) µ(1 + p 2, 1) = µ(p 1 + rp 2, r). (35) We have also µ(1 + p, 1) t = µ(1 + tp, 1). (36) The Fuss numbers ( ) np+1 1 n np+1 are moments of MP p 1, p 1. Wojciech M lotkowski Fuss-Catalan numbers / 35

27 Boolean convolution, Bożejko, Speicher, Woroudi The Boolean convolution µ 1 µ 2 and the Boolean power µ t can be defined by putting 1 M µ1 µ 2 (z) = 1 M µ1 (z) + 1 1, (37) M µ2 (z) M µ t (z) = the latter is well defined for all t > 0. From the formula B p (z) 1+r (1 p)b p (z) + p = M µ (z) (1 t)m µ (z) + t, (38) n=0 ( np + r n ) z n. (39) we see that for p 1 the numbers ( np n ) are moments of µ(p, 1) p. Wojciech M lotkowski Fuss-Catalan numbers / 35

28 A by-product: Consider the dilation D 1/p µ(p, 1) p. Its moments are ( np n ) 1 p n. It is easy to check that ( np n ) 1 p n 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 free multiplicative convolution, Studia Math. 214 (2013), Wojciech M lotkowski Fuss-Catalan numbers / 35

29 Monotonic convolution, Muraki 2000 The Monotonic convolution is an associative, noncommuting operation µ 1 µ 2 = µ on probability measures on R which is defined by M µ (z) = M µ1 ( zmµ2 (z) ) M µ2 (z). (41) Then the formula B p (z) = B p r ( zbp (z) r ) leads to: a 1, 0 b a, r > 0. Corollary µ(a, b) µ(a + r, r) = µ(a + r, b + r), (42) )} m=0 If 0 r p 1 then the sequence {( mp+r m is positive definite as the moment sequence of µ(p r, 1) p µ(p, r). Wojciech M lotkowski Fuss-Catalan numbers / 35

30 M lotkowski, Penson, Probability distributions with binomial moments, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 17/2, Wojciech M lotkowski Fuss-Catalan numbers / 35 Theorem The sequence {( mp+r m 1 r p 1. )} m=0 is positive definite if and only if p 1 and

31 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 µ(p, r). For c > 0 define probability measure η(c) by with moments { } c n+c. n=0 From the formula ( np + r 1 n η(c) := c x c 1 dx, x [0, 1]. ) we get Mellin convolution relation: Theorem For p > 1, 0 < r p we have r n(p 1) + r = ( np + r ν(p, r 1) η (r/(p 1)) = µ(p, r). n ) r np + r Wojciech M lotkowski Fuss-Catalan numbers / 35

32 Theorem For p > 1 we have ( 1 ν(p, 1) = D c(p) p δ 0 + p 1 ) p p δ 1, (43) where c(p) = p p (p 1) 1 p, and ( ( 1 ν(p, 0) = D p p δ 0 + p 1 ) ) p/(p 1) p 1 p δ 1. (44) D denotes dilation: D c (µ)(e) := µ ( 1 c E). Wojciech M lotkowski Fuss-Catalan numbers / 35

33 Another family of moment sequences For p, t > 0 define ( np + 1 a n (p, t) := n ( ) np = n ) t np ( np + 2 n(2p t pt) + 2 (np n + 1)(np n + 2), From the implicit formula for the weight function W p,r : n ) 2(1 t) np + 2 W p,r (f p (φ)) = sin φ sin rφ (sin(p 1)φ)p r 1 π (sin pφ) p r, we can prove f p (φ) = (sin pφ) p sin φ (sin(p 1)φ) p 1, 0 < φ < π/p, Wojciech M lotkowski Fuss-Catalan numbers / 35

34 Theorem The sequence a n (p, t) is positive definite if and only if p 1 and where g(p) t 2p 1 + p, g(p) = min{t R : t sin(φ φ p ) + 2(1 t) sin φ cos φ p 0 for 0 < φ < π}. The case t = 2p 1+p appears in M. Bousquet-Mélou and G. Schaeffer, Enumeration of planar constellations, Adv. in Appl. Math. 24 (2000), Wojciech M lotkowski Fuss-Catalan numbers / 35

35 1.5 2 p p g p p Wojciech M lotkowski Fuss-Catalan numbers / 35