PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 139, Number 10, October 2011, Pages 3735–3738
S 0002-9939(2011)11015-3
Article electronically published on May 24, 2011
ON EXPLICIT PROBABILITY DENSITIES
ASSOCIATED WITH FUSS-CATALAN NUMBERS
DANG-ZHENG LIU, CHUNWEI SONG, AND ZHENG-DONG WANG
(Communicated by Jim Haglund)
Abstract. In this paper we give explicitly a family of probability densities,
the moments of which are Fuss-Catalan numbers. The densities appear naturally in random matrices, free probability and other contexts.
In this paper we study a family of probability densities πs , s ∈ N, which are
uniquely determined by the moment sequences {m0 , m1 , . . . , mk , . . . } [1]. Here
1
sk + k
(0.1)
mk =
k
sk + 1
are known as Fuss-Catalan numbers in free probability theory [7]. The densities πs
belong to the class of free Bessel laws [3] and are known to appear in several different
contexts, for instance, random matrices [1, 3, 7], random quantum states [4], free
probability and quantum groups [3, 6]. More precisely, πs is the limit spectral distribution of random matrices in the forms such as X1s X1∗s and X1 · · · Xs Xs∗ · · · X1∗ ,
where X1 , . . . , Xs are independent N × N random matrices (the matrix elements of
different matrices are independent); in free probability we have the free convolution
relation: πs = π1s [7, 3].
More generally, T. Banica et al. [3] introduce a remarkable family of probability
distributions πs,t with (s, t) ∈ (0, ∞) × (0, ∞) − (0, 1) × (1, ∞), called free Bessel
laws. πs is the special case where t = 1, i.e., πs,1 = πs . The moments of πs,t are
called the Fuss-Catalan polynomials (Theorem 5.2, [3]):
k
1 k−1
sk
tj .
(0.2)
mk (t) =
j−1
j−1
j
j=1
Indeed, the following relation holds [3]:
(0.3)
πs,1 = π1s ,
π1,t = π1t .
The distribution π1,t , the famous Marchenko-Pastur law of parameter t, also
called free Poisson law, has an explicit formula:
4t − (x − 1 − t)2
(0.4)
π1,t = max(1 − t, 0)δ0 +
.
2πx
Received by the editors August 4, 2010.
2010 Mathematics Subject Classification. Primary 60B20, 46L54; Secondary 05A99.
Key words and phrases. Fuss-Catalan numbers, moments, free Bessel laws.
c
2011
American Mathematical Society
3735
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3736
DANG-ZHENG LIU, CHUNWEI SONG, AND ZHENG-DONG WANG
In the special case,
1 −1
4x − 1.
2π
where an explicit density formula is available is due to
π1 = π1,1 =
(0.5)
Another special case of πs,t
Penson and Solomon [8]:
√
√ √
√
√
3
2 3 3 2(27 + 3 81 − 12x)2/3 − 6 3 x
√
1(0,27/4] (x).
(0.6)
π2 = π2,1 =
12π
x2/3 (27 + 3 81 − 12x)1/3
To the best of our knowledge, except for the special cases above there are no
explicit formulae available for the other πs,t . The aim of this work is to give explicit
densities of πs = πs,1 , s ∈ N. The proof of the following result is based on the
method to find an explicit density from a given moment sequence used in [9, 5].
Theorem 0.1. Let πs , s ∈ N, be the unique densities determined by the FussCatalan numbers in Eq. (0.1). Then we have the following formulae:
1/s−1/2
1(0,K] (x)
τK − x
(0.7)
πs (x) =
F (t1 , . . . , ts ) 1{τ K≥x} ds t,
B( 12 , 12 + 1s ) [0,1]s √x τ K 1/s
where K = (s + 1)s+1 /ss , τ =
s
tj and F (t1 ) = δ(t1 − 1), while for s > 1
j=1
1
F (t1 , . . . , ts ) =
1
s−1
B( s+1
, 2s+2
)
s
B(
j=2
(0.8)
1
s+1 −1
× t1
j
j
,
)
s + 1 s(s + 1)
(1 − t1 ) 2s+2 −1
s−1
s
j
tjs+1
−1
j
(1 − tj ) s(s+1) −1 .
j=2
Proof. First, we derive a family of symmetric distributions σs , the 2k-moments of
which are mk in Eq. (0.1).
Consider the characteristic function of σs as follows:
+∞
∞
∞
(−ξ 2 )k
(−ξ 2 )k
(0.9)
mk =
,
eiξx σs (x)dx =
βk
(2k)!
k!
−∞
k=0
where
(0.10)
1
βk =
sk + 1
k=0
k!
1
(sk + k)!
sk + k
=
.
k
(2k)!
sk + 1 (sk)!(2k)!
A direct computation shows that the ratio
βk+1
1
sk + 1
=
βk
s(k + 1) + 1 (2k + 1)(2k + 2)
(s + 1)k + 1 (s + 1)k + 2 · · · (s + 1)k + s + 1
×
(sk + 1)(sk + 2) · · · (sk + s)
1
2
s
)(k
+ s+1
) · · · (k + s+1
)
(k
+
1
K
s+1
=
1
2
s
4
(k + 2 )(k + s ) · · · (k + s )
k + 1 + 1s
1
. K (k + a1 )(k + a2 ) · · · (k + as )
=
.
4 (k + b1 )(k + b2 ) · · · (k + bs ) k + bs+1
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PROBABILITY DENSITIES AND FUSS-CATALAN NUMBERS
3737
Here
1
i
i+1
, ai =
and bi+1 =
for i = 1, 2, . . . , s.
2
s+1
s
Therefore, using the generalized hypergeometric function, we rewrite
+∞
K
(0.12)
eiξx σs (x)dx = s Fs+1 (a1 , . . . , as ; b1 , . . . , bs , bs+1 ; − ξ 2 )
4
−∞
τK 2 s
ξ ) d t.
(0.13)
F (t1 , . . . , ts ) 0 F1 (bs+1 ; −
=
4
[0,1]s
(0.11)
b1 =
Note that a1 = b1 for s = 1 but bj > aj > 0, j = 1, 2, . . . , s, when s > 1. In
Eq. (0.13) we have made use of Euler’s integral representation of the generalized
hypergeometric function [2].
Next, with the help of the integral representation of the Bessel function of the
first kind [2], that is, for α > − 12 ,
(z/2)α
1 2
0 F1 (α + 1; − z )
Γ(α + 1)
4
1
α
1
(z/2)
=√
eizx (1 − x2 )α− 2 dx,
πΓ(α + 12 ) −1
Jα (z) =
(0.14)
we get
1 2
Γ(α + 1)
0 F1 (α + 1; − z ) = √
4
πΓ(α + 12 )
1
=
1
B( 2 , α + 12 )
(0.15)
√
Set α = 1/s, z = τ Kξ. We have
0 F1 (bs+1 ; −
τK 2
1
ξ )=
1
4
B( 2 , 12 + 1s )
(0.16)
=
1
B( 12 , 12 + 1s )
1
1
eizx (1 − x2 )α− 2 dx
−1
1
1
eizx (1 − x2 )α− 2 dx.
−1
1
ei
√
τ Kξx
−1
√
τK
√
− τK
(1 − x2 ) s − 2 dx
1
1
(τ K − x2 ) s − 2
1
eiξx
1
(τ K) s
1
dx.
Combining (0.16) and (0.13) after interchanging the order of integration, we
obtain
1/s−1/2
τ K − x2
1
(0.17) σs (x) =
F (t1 , . . . , ts ) 1{τ K≥x2 } ds t.
1/s
B( 12 , 12 + 1s ) [0,1]s
τK
Noting the fact that
(0.18)
πs (x) =
√
σs ( x)
√
1(0,∞) ,
x
the proof is then complete.
We remark that some properties of the density πs (x) follow easily from Theorem 0.1; for instance, there are no atoms, the support is [0, K], and the density
is analytic on (0, K) (these properties were obtained earlier by implicit complexanalytic methods; see Theorem 2.1, [3]).
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3738
DANG-ZHENG LIU, CHUNWEI SONG, AND ZHENG-DONG WANG
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School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
Current address: Instituto Matemática y Fı́sica, Universidad de Talca, Casilla 747, Talca,
Chile
E-mail address: [email protected]
School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
E-mail address: [email protected]
School of Mathematical Sciences, LMAM, Peking University, Beijing, 100871, People’s Republic of China
E-mail address: [email protected]
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