Densities of the Raney Distributions

Wojciech Młotkowski^{1}, Karol A. Penson^{2}, Karol Życzkowski^{3}

Received: October 7, 2012 Revised: November 2, 2013

Communicated by Friedrich Götze

Abstract. We prove that if p ≥ 1 and 0 < r ≤ p then the sequence \binom{mp+r}{m} \frac{r}{mp+r} is positive definite. More precisely, it is the moment sequence of a probability measure μ(p, r) with compact support contained in [0, +∞). This family of measures encompasses the multiplicative free powers of the Marchenko-Pastur distribution as well as the Wigner’s semicircle distribution centered at x = 2. We show that if p > 1 is a rational number and 0 < r ≤ p then μ(p, r) is absolutely continuous and its density W_{p,r}(x) can be expressed in terms of the generalized hypergeometric functions. In some cases, including the multiplicative free square and the multiplicative free square root of the Marchenko-Pastur measure, W_{p,r}(x) turns out to be an elementary function.

2010 Mathematics Subject Classification: Primary 44A60; Secondary 33C20 Keywords and Phrases: Mellin convolution, free convolution, Meijer G-function, generalized hypergeometric function.

1W. M. is supported by the Polish National Science Center grant No. 2012/05/B/ST1/ 00626.

2K. A. P. acknowledges support from PAN/CNRS under Project PICS No. 4339 and from Agence Nationale de la Recherche (Paris, France) under Program PHYSCOMB No.

ANR-08-BLAN-0243-2.

3K. Ż. is supported by the Grant DEC-2011/02/A/ST1/00119 of Polish National Centre of Science.

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Introduction

For p, r ∈ ℝ we define the Raney numbers (or two-parameter Fuss-Catalan numbers) by

A_{m}(p, r) := \frac{r}{m!} ∏_{i=1}^{m−1}(mp + r − i),  (1)

A_{0}(p, r) := 1. We can also write

A_{m}(p, r) = \binom{mp + r}{m}\frac{r}{mp + r} ,  (2)

(unless mp + r = 0), where the generalized binomial is defined by

\binom{a}{m} := \frac{a(a − 1) … (a − m + 1)}{m!}.

Let B_{p}(z) denote the generating function of the sequence {A_{m}(p, 1)}_{m=0}^{∞}, the Fuss numbers of order p:

B_{p}(z) := ∑_{m=0}^{∞} A_{m}(p, 1)z^{m},  (3)

convergent in some neighborhood of 0. For example

B_{2}(z) = \frac{2}{1 + \sqrt{1 − 4z}} .  (4)

Lambert showed that

B_{p}(z)^{r} = ∑_{m=0}^{∞} A_{m}(p, r)z^{m},  (5)

see [9]. These generating functions also satisfy

B_{p}(z) = 1 + zB_{p}(z)^{p},  (6)

which reflects the identity A_{m}(p, p) = A_{m+1}(p, 1), and

B_{p}(z) = B_{p−r}(zB_{p}(z)^{r}).  (7)

Using the free probability theory (see [28, 18, 6]) it was shown in [16] that if p ≥ 1 and 0 ≤ r ≤ p then the sequence {A_{m}(p, r)}_{m=0}^{∞} is positive definite, i.e. is the moment sequence of a probability measure μ(p, r) on ℝ. Moreover, μ(p, r) has compact support (and therefore is unique) contained in the positive half-line [0, ∞) (for example μ(p, 0) = δ_{0}). The measures μ(p, r) satisfy some interesting relations, for example

μ(p_{1}, r) ⊠ μ(1 + p_{2}, 1) = μ(p_{1} + rp_{2}, r)  (8)

and

μ(p, r) ▷ μ(p + s, s) = μ(p + s, r + s),  (9)

see [16], where “⊠” and “▷” denotes the multiplicative free and the monotonic convolution (see [17]). A relation analogous to (9) is also satisfied by the threeparameter family of distributions studied by Arizmendi and Hasebe [4].

Among the measures μ(p, r) perhaps the most important is the Marchenko-Pastur (called also the free Poisson) distribution

μ(2, 1) = \frac{1}{2π} \sqrt{\frac{4 − x}{x}} dx  on [0,4],  (10)

which plays an important role in the theory of random matrices, see [29, 10, 11, 2, 1, 5]. It was proved in [1] that the multiplicative free power μ(2, 1)^{⊠n} = μ(n+1, 1) is the limit of the distribution of squared singular values of the power G^{n} of a random matrix G, when the size of the matrix G goes to infinity. The moments of μ(2, 1), A_{m}(2, 1) = \binom{2m+1}{m}/(2m + 1), are called Catalan numbers and play an important role in combinatorics, see A000108 in OEIS [24].

In this paper we are going to prove positive definiteness of {A_{m}(p, r)}_{m=0}^{∞} using more classical methods. Namely, we show that if p > 1, 0 < r ≤ p and if p is a rational number then μ(p, r) is absolutely continuous and can be represented as Mellin convolution of modified beta measures. Next we provide a formula for the density W_{p,r}(x) of μ(p, r) in terms of the Meijer G-function and of the generalized hypergeometric functions (cf. [30, 21], where p was assumed to be an integer). This allows us to draw graphs of these densities and, in some particular cases, to express W_{p,r}(x) as an elementary function.

Let us mention that the measures μ(2, 1)^{⊠p} = μ(1 + p, 1) were also studied by Banica, Belinschi, Capitaine and Collins [5] as a special case of the free Bessel laws. They showed in particular that for p > 0 this measure is absolutely continuous and its support is [0, (p + 1)^{p+1}p^{−p}]. Liu, Song and Wang [14] found a formula expressing the density of μ(2, 1)^{⊠n}, n natural, as integral of a certain kernel over [0, 1]^{n}. Recently Haagerup and Möller [12] studied a twoparameter family μ_{α,β}, α, β > 0, of probability measures. The measures μ_{α,0} coincide with our μ(1 + α, 1), but if β > 0 then μ_{α,β} has noncompact support, so it does not coincide with any of μ(p, r). The authors found a formula for the density function of μ_{α,β}, which in the case of W_{1+p,1} reads as follows:

W_{1+p,1} (\frac{sin^{p+1}((p + 1)t)}{sin t sin^{p}(pt)}) = \frac{sin^{2} t sin^{p−1}(pt)}{π sin^{p}((p + 1)t)},  (11)

for 0 < t < π/(p + 1). It can be used for drawing the graph of W_{1+p,1}(x) by computer.

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1 Preliminaries

For probability measures μ_{1}, μ_{2} on the positive half-line [0, ∞) the Mellin convolution is defined by

(μ_{1} ∘ μ_{2}) (A) := ∫_{0}^{∞} ∫_{0}^{∞} 1_{A}(xy)dμ_{1}(x)dμ_{2}(y)  (12)

for every Borel set A ⊆ [0, ∞). This is the distribution of product X_{1} · X_{2} of two independent nonnegative random variables with X_{i} ∼ μ_{i}. In particular, μ ∘ δ_{c} (c > 0) is the dilation of μ:

(μ ∘ δ_{c}) (A) = D_{c}μ(A) := μ (\frac{1}{c}A) .

If μ has density f(x) then D_{c}(μ) has density f(x/c)/c. If both the measures μ_{1}, μ_{2} have all moments

s_{m}(μ_{i}) := ∫_{0}^{∞} x^{m} dμ_{i}(x)

finite then so has μ_{1} ∘ μ_{2} and

s_{m} (μ_{1} ∘ μ_{2}) = s_{m}(μ_{1}) · s_{m}(μ_{2})

for all m. If μ_{1}, μ_{2} are absolutely continuous, with densities f_{1}, f_{2} respectively, then so is μ_{1} ∘ μ_{2} and its density is given by the Mellin convolution:

(f_{1} ∘ f_{2}) (x) := ∫_{0}^{∞} f_{1}(x/y)f_{2}(y)\frac{dy}{y} .

We will need the following modified beta measures:

Lemma 1.1. Let u, v, l > 0. Then

\{\frac{Γ(u + n/l)Γ(u + v)}{Γ(u + v + n/l)Γ(u)}\}_{n=0}^{∞}

is the moment sequence of the probability measure

b(u + v, u, l) := \frac{l}{B(u, v)}x^{lu−1} (1 − x^{l})^{v−1} dx  (13)

on [0, 1], where B is the Euler beta function.

Proof. Using the substitution t = x^{l} we obtain:

\frac{Γ(u + n/l)Γ(u + v)}{Γ(u + v + n/l)Γ(u)} = \frac{B(u + n/l, v)}{B(u, v)} = \frac{1}{B(u, v)} ∫_{0}^{1} t^{u+n/l−1}(1 − t)^{v−1}dt

= \frac{l}{B(u, v)} ∫_{0}^{1} x^{lu+n−1} (1 − x^{l})^{v−1} dx.

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Note that if X is a positive random variable whose distribution has density f(x) and if l > 0 then the distribution of X^{1/l} has density lx^{l−1}f(x^{l}). In particular, if the distribution of a random variable X is b(u + v, u, 1) then the distribution of X^{1/l} is b(u + v, u, l). For u, l > 0 we also define

b(u, u, l) := δ_{1}.  (14)

2 Applying Mellin convolution

From now on we assume that p > 1 is a rational number, say p = k/l, with 1 ≤ l < k, and that 0 < r ≤ p. We will show that then A_{m}(p, r) is the moment sequence of a probability measure μ(p, r), which can be represented as Mellin convolution of modified beta measures. In particular, μ(p, r) is absolutely continuous and we will denote its density by W_{p,r}. The case when p is an integer was studied in [21, 30]. First we need to express the numbers A_{m}(p, r) in a special form.

Lemma 2.1. If p = k/l, where k, l are integers, 1 ≤ l < k and 0 < r ≤ p then

A_{m}(p, r) = \frac{r}{l\sqrt{2πk(p − 1)}} (\frac{p}{p − 1})^{r} \frac{∏_{j=1}^{k} Γ(β_{j} + m/l)}{∏_{j=1}^{k} Γ(α_{j} + m/l)}c(p)^{m}, (15)

where c(p) = p^{p}(p − 1)^{1−p},

α_{j} = \{ \begin{matrix}\frac{j}{l} & if 1 ≤ j ≤ l, \\ \frac{r + j − l}{k − l} & if l + 1 ≤ j ≤ k, \end{matrix}  (16)

β_{j} = \frac{r + j − 1}{k},  1 ≤ j ≤ k.  (17)

Proof. First we write:

\binom{mp + r}{m}\frac{r}{mp + r} = \frac{rΓ(mp + r)}{Γ(m + 1)Γ(mp − m + r + 1)}.  (18)

Now we apply the Gauss’s multiplication formula:

Γ(nz) = (2π)^{(1−n)/2}n^{nz−1/2}Γ(z)Γ (z + \frac{1}{n}) Γ (z + \frac{2}{n}) … Γ (z + \frac{n − 1}{n})

to get:

Γ(mp + r) = Γ (k (\frac{m}{l} + \frac{r}{k} ))

= (2π)^{(1−k)/2}k^{mk/l+r−1/2} ∏_{j=1}^{k} Γ (\frac{m}{l} + \frac{r + j − 1}{k}) ,

Γ(m + 1) = Γ (l \frac{m + 1}{l}) = (2π)^{(1−l)/2}l^{m+1/2} ∏_{j=1}^{l} Γ (\frac{m}{l} + \frac{j}{l} )

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1578 W. Młotkowski, K. A. Penson, K. Życzkowski and

Γ(mp − m + r + 1) = Γ ((k − l) (\frac{m}{l} + \frac{r + 1}{k − l}))
= (2π)^{(1−k+l)/2}(k − l)^{m(k−l)/l+r+1/2} ∏_{j=l+1}^{k} Γ (\frac{m}{l} + \frac{r + j − l}{k − l} ) .

It remains to use them in (18). In order to apply Lemma 1.1 we need to modify enumeration of α’s. Lemma 2.2. For 1 ≤ i ≤ l + 1 denote

j_{i} := ⌊\frac{(i − 1)k}{l} ⌋ + 1,

where ⌊·⌋ is the floor function, so that

1 = j_{1} < j_{2} < … < j_{l} < k < k + 1 = j_{l+1}.

For 1 ≤ j ≤ k define

α̃_{j} = \{ \begin{matrix}\frac{i}{l} & if j = j_{i}, 1 ≤ i ≤ l, \\ \frac{r + j − i}{k − l} & if j_{i} < j < j_{i+1}. \end{matrix}  (19)

Then the sequence \left\{{ \atop α_{j}}\right\}_{j=1}^{k} is a rearrangement of \{α_{j}\}_{j=1}^{k}. Moreover, if 0 < r ≤ p = k/l then we have β_{j} ≤ α̃_{j} for all j ≤ k. Proof. It is easy to verify the first statement. Assume that j = j_{i} for some i ≤ l. We have to show that

\frac{r + j_{i} − 1}{k} ≤ \frac{i}{l},

which is equivalent to

lr + l ⌊\frac{k(i − 1)}{l}⌋ ≤ ki.

The latter is a consequence of the fact that ⌊x⌋ ≤ x and the assumption that r ≤ p = k/l. Now assume that j_{i} < j < j_{i+1}. We ought to show that

\frac{r + j − 1}{k} ≤ \frac{r + j − i}{k − l},

which is equivalent to

lr + lj + k − l − ki ≥ 0.

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Using the inequality ⌊x⌋ + 1 > x we obtain

lj + k − l − ki ≥ l(j_{i} + 1) + k − l − ki
= lj_{i} + k − ki > k(i − 1) + k − ki = 0,

which completes the proof, as r > 0.

Now we are ready to prove the main theorem of this section.

Theorem 2.3. Suppose that p = k/l, where k, l are integers, 1 ≤ l < k, and that r is a real number such that 0 < r ≤ p. Then there exists a unique probability measure μ(p, r) such that (1) is its moment sequence. Moreover μ(p, r) can be represented as the following Mellin convolution:

μ(p, r) = b(α̃_{1}, β_{1}, l) ∘ … ∘ b(α̃_{k}, β_{k}, l) ∘ δ_{c(p)},

where

c(p) := \frac{p^{p}}{(p − 1)^{p−1}} .

Consequently, μ(p, r) is absolutely continuous and its support is [0, c(p)].

It is easy to see that the density function is positive on (0, c(p)). The representation of densities in the form of Mellin convolution of modified beta measures was used in different context in [8], see its Appendix A. Example. For the Marchenko-Pastur measure we get the following decomposition:

μ(2, 1) = b(1, 1/2, 1) ∘ b(2, 1, 1) ∘ δ_{4},  (20)

where b(1, 1/2, 1) has density 1/(π\sqrt{x − x^{2}}) on [0, 1], the arcsine distribution with the moment sequence \binom{2m}{m}4^{−m}, and b(2, 1, 1) is the Lebesgue measure on [0, 1] with the moment sequence 1/(m + 1).

Proof. In view of Lemma 2.1 and Lemma 2.2 we can write

A_{m}(p, r) = D ∏_{j=1}^{k} \frac{Γ(β_{j} + m/l)Γ(α̃_{j})}{Γ(α̃_{j} + m/l)Γ(β_{j})} · c(p)^{m}

for some constant D. Taking m = 0 we see that D = 1.

Note that a part of the theorem illustrates a result of Kargin [13], who proved that if μ is a compactly supported probability measure on [0, ∞), with expectation 1 and variance V, and if L_{n} denotes the supremum of the support of the multiplicative free convolution power μ^{⊠n}, then

lim_{n→∞} \frac{L_{n}}{n} = eV,  (21)

where e = 2.71 … is the Euler’s number. The Marchenko-Pastur measure μ(2, 1) has expectation and variance equal to 1 and μ(2, 1)^{⊠n} = μ(n + 1, 1), so in this case L_{n} = (n+1)^{n+1}/n^{n} (this was also proved in [29] and [11]) and (21) holds. The density function for μ(p, r) will be denoted by W_{p,r}(x). Since A_{m}(p, p) = A_{m+1}(p, 1), we have

W_{p,p}(x) = x · W_{p,1}(x),  (22)

for example

W_{2,2}(x) = \frac{1}{2π} \sqrt{x(4 − x)}  on [0, 4],  (23)

which is the semicircle Wigner distribution with radius 2, centered at x = 2. Now we can reprove the main result of [16].

Theorem 2.4. Suppose that p, r are real numbers satisfying p ≥ 1, 0 ≤ r ≤ p. Then there exists a unique probability measure μ(p, r), with compact support contained in [0, c(p)], such that {A_{m}(p, r)}_{m=0}^{∞} is its moment sequence.

Proof. It follows from the fact that the class of positive definite sequence is closed under pointwise limits.

Remark. In view of Theorem 2.1 in [5], for every p > 1 the measure μ(p, 1) is absolutely continuous and its support is equal [0, c(p)], see also [14, 12].

3 Applying Meijer G-function

The aim of this section is to describe the density function W_{p,r}(x) of μ(p, r) in terms of the Meijer G-function (see [19] for example) and consequently, as a linear combination of generalized hypergeometric functions. We will see that in some particular cases W_{p,r} can be represented as an elementary function. For p > 1, r > 0 define an analytic function

ϕ_{p,r}(σ) = \frac{rΓ((σ − 1)p + r)}{Γ(σ)Γ((σ − 1)(p − 1) + r + 1)},

which is well defined whenever (σ − 1)p + r is not a nonpositive integer. Note that ϕ_{p,1}(σ + 1) = ϕ_{p,p}(σ) and if m is a natural number then

ϕ_{p,r}(m + 1) = \binom{mp + r}{m}\frac{r}{mp + r} .

Then we define W_{p,r} as the inverse Mellin transform:

W_{p,r}(x) = \frac{1}{2πi} ∫_{d−i∞}^{d+i∞} x^{−σ}ϕ_{p,r}(σ) dσ,

x > 0, if exists, see [25] for details. It turn out that if p > 1 is a rational number then W_{p,r} can be expressed in terms of the Meijer G-function and its Mellin transform is ϕ_{p,r}. For the theory of the Meijer G-functions we refer to [15, 23, 19].

Theorem 3.1. Suppose that p = k/l, where k, l are integers, 1 ≤ l < k and r > 0. Then W_{p,r}(x) is well defined and

W_{p,r}(x) = \frac{rp^{r}}{x(p − 1)^{r+1/2}\sqrt{2kπ}} G_{k,k}^{k,0}(\frac{x^{l}}{c(p)^{l}} | \begin{matrix}α_{1}, & …, & α_{k} \\ β_{1}, & …, & β_{k} \end{matrix} ) ,  (24)

x ∈ (0, c(p)), where c(p) = p^{p}(p − 1)^{1−p} and the parameters α_{j}, β_{j} are given by (16) and (17). Moreover, ϕ_{p,r} is the Mellin transform of W_{p,r}, namely

ϕ_{p,r}(σ) = ∫_{0}^{c(p)} x^{σ−1}W_{p,r}(x) dx,  (25)

for ℜσ > 1 − r/p. If 0 < r ≤ p then W_{p,r}(x) > 0 for 0 < x < c(p) and therefore W_{p,r} is the density function of the probability distribution μ(p, r).

Proof. Putting m = σ − 1 in (15) we get

ϕ_{p,r}(σ) = \frac{r(p − 1)^{p−r−3/2}}{lp^{p−r}\sqrt{2kπ}} \frac{∏_{j=1}^{k} Γ(β_{j} − 1/l + σ/l)}{∏_{j=1}^{k} Γ(α_{j} − 1/l + σ/l)}c(p)^{σ}.  (26)

Writing the right hand side as Φ(σ/l − 1/l)c(p)^{σ}, using the substitution σ = lu + 1 and the definition of the Meijer G-function (see [19] for example), we obtain

W_{p,r}(x) = \frac{1}{2πi} ∫_{d−i∞}^{d+i∞} Φ(σ/l − 1/l)c(p)^{σ}x^{−σ}dσ

= \frac{lc(p)}{2πxi} ∫_{d−i∞}^{d+i∞} Φ(u) (x^{l}/c(p)^{l})^{−u} du

= \frac{rp^{r}}{x(p − 1)^{r+1/2}\sqrt{2kπ}} G_{k,k}^{k,0}(z | \begin{matrix}α_{1}, & …, & α_{k} \\ β_{1}, & …, & β_{k} \end{matrix} ) ,

where z = x^{l}/c(p)^{l}. Recall that for the Meijer function of type G_{k,k}^{k,0} there is no restriction on the parameters and the integral converges for 0 < x < c(p) (see 16.17.1 in [19]). On the other hand, substituting x = c(p)t^{1/l} we can write

∫_{0}^{c(p)}x^{σ−1}W_{p,r}(x) dx

= \frac{rp^{r}}{(p − 1)^{r+1/2}\sqrt{2kπ}} ∫_{0}^{c(p)} x^{σ−2}G_{k,k}^{k,0}(\frac{x^{l}}{c(p)^{l}} | \begin{matrix}α_{1}, & …, & α_{k} \\ β_{1}, & …, & β_{k} \end{matrix} ) dx.

= \frac{rp^{r}c(p)^{σ−1}}{l(p − 1)^{r+1/2}\sqrt{2kπ}} ∫_{0}^{1} t^{(σ−1)/l−1}G_{k,k}^{k,0}(t | \begin{matrix}α_{1}, & …, & α_{k} \\ β_{1}, & …, & β_{k} \end{matrix} ) dt.

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Since ∑_{j=1}^{k} (β_{j} − α_{j}) = −3/2 < 0, so the assumptions of (2.24.2.1) in [23], the third case, are satisfied and therefore the last integral is convergent provided

−\frac{r}{k} = − min β_{j} < ℜ\frac{σ − 1}{l},

(equivalently: ℜσ > 1 − r/p) and the whole expression is equal to the right hand side of (26). For the last statement we note that in view of Theorem 2.3, of the uniqueness part of the Riesz representation theorem for linear functionals on C[0, c(p)] and of the Weierstrass approximation theorem, for 0 < r ≤ p the density function of μ(p, r) must coincide with W_{p,r}.

Now applying Slater’s formula we can express W_{p,r} as a linear combination of hypergeometric functions.

Theorem 3.2. For p = k/l, with 1 ≤ l < k, r > 0, and x ∈ (0, c(p)) we have

W_{p,r}(x) = γ(k, l, r) ∑_{h=1}^{k} c(h, k, l, r)_{k}F_{k−1}(\begin{matrix}a(h, k, l, r) \\ b(h, k, l, r) \end{matrix} | z) z^{(r+h−1)/k−1/l},

(27)

where z = x^{l}/c(p)^{l},

γ(k, l, r) = \frac{r(p − 1)^{p−r−3/2}}{p^{p−r}\sqrt{2kπ}} ,  (28)

c(h, k, l, r) = \frac{∏_{j=1}^{h−1} Γ (\frac{j−h}{k}) ∏_{j=h+1}^{k} Γ (\frac{j−h}{k})}{∏_{j=1}^{l} Γ (\frac{j}{l} − \frac{r+h−1}{k}) ∏_{j=l+1}^{k} Γ (\frac{r+j−l}{k−l} − \frac{r+h−1}{k})},  (29)

and the parameter vectors of the hypergeometric functions are

a(h, k, l, r) = (\{\frac{r + h − 1}{k} − \frac{j − l}{l} \}_{j=1}^{l} , \{\frac{r + h − 1}{k} − \frac{r + j − k}{k − l} \}_{j=l+1}^{k}) , (30)

b(h, k, l, r) = (\{\frac{k + h − j}{k} \}_{j=1}^{h−1} , \{\frac{k + h − j}{k} \}_{j=h+1}^{k}) .  (31)

Proof. Putting z = x^{l}/c(p)^{l}, and hence x = c(p)z^{1/l}, we can rewrite (24) as

W_{p,r}(x) = \frac{r(p − 1)^{p−r−3/2}}{z^{1/l}p^{p−r}\sqrt{2kπ}} G_{k,k}^{k,0}(z | \begin{matrix}α_{1}, & …, & α_{k} \\ β_{1}, & …, & β_{k} \end{matrix} ) ,  (32)

x ∈ (0, c(p)). Observe that for 1 ≤ i < j ≤ k the difference β_{j} − β_{i} = (j − i)/k is never an integer. Therefore we can apply formula (8.2.2.3) in [23] (see also (16.17.2) in [19] or formula (7) on page 145 in [15]), so that

c(h, k, l, r) = \frac{∏_{j≠h} Γ(β_{j} − β_{h})}{∏_{j=1}^{k} Γ(α_{j} − β_{h})},

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which gives (29). For the parameter vectors we have

a(h, k, l, r)_{j} = 1 + β_{h} − α_{j}

and

b(h, k, l, r)_{j} = 1 + β_{h} − β_{j},  j ≠ h,

which leads to (30) and (31). Finally, the summand with index h is in addition multiplied by z^{β_{h}−1/l}.

Theorem 3.1 and Theorem 3.2 are sufficient for drawing graphs of the functions W_{p,r} with help of computer programs. In some cases however it is possible to express W_{p,r} as an elementary function. The most tractable case is p = 2. We know already that

W_{2,1}(x) = \frac{1}{2π} \sqrt{\frac{4 − x}{x}}, W_{2,2}(x) = \frac{1}{2π} \sqrt{x(4 − x)}.

Now we can give a simple formula for W_{2,r}.

Corollary 3.3. For p = 2, r > 0, the function W_{2,r} is

W_{2,r}(x) = \frac{sin (r · arccos \sqrt{x/4})}{πx^{1−r/2}},  (33)

x ∈ (0, 4). If 0 < r ≤ 2 then W_{2,r} is the density function of the measure μ(2, r). In particular for r = 1/2 and r = 3/2 we have

W_{2,1/2}(x) = \frac{\sqrt{2 − \sqrt{x}}}{2πx^{3/4}},  (34)
W_{2,3/2}(x) = \frac{(\sqrt{x} + 1) \sqrt{2 − \sqrt{x}}}{2πx^{1/4}}.  (35)

Note that if r > 2 then W_{2,r}(x) < 0 for some values of x ∈ (0, 4).

Proof. We take k = 2, l = 1 so that c(2) = 4, z = x/4 and γ(2, 1, r) = r2^{r}/(8\sqrt{π}). Using the Euler’s reflection formula and the identity Γ(1 + r/2) = Γ(r/2)r/2 we get

c(1, 2, 1, r) = \frac{Γ(1/2)}{Γ(1 − r/2)Γ(1 + r/2)} = \frac{2 sin(πr/2)}{r\sqrt{π}},
c(2, 2, 1, r) = \frac{Γ(−1/2)}{Γ((1 − r)/2)Γ((1 + r)/2)} = \frac{−2 cos(πr/2)}{\sqrt{π}}.

Documenta Mathematica 18 (2013) 1573–1596

We also need formulas for two hypergeometric functions, namely

2F_{1}( \frac{r}{2}, \frac{−r}{2} ; \frac{1}{2} | z) = cos(r arcsin \sqrt{z}),
2F_{1}( \frac{1 + r}{2} , \frac{1 − r}{2} ; \frac{3}{2} | z) = \frac{sin(r arcsin \sqrt{z})}{r\sqrt{z}},

see 15.4.12 and 15.4.16 in [19]. Now we can write

W_{2,r}(x) = \frac{sin(πr/2) cos (r arcsin \sqrt{x/4}) − cos(πr/2) sin (r arcsin \sqrt{x/4})}{πx^{1−r/2}}

= \frac{sin (πr/2 − r arcsin \sqrt{x/4})}{πx^{1−r/2}} = \frac{sin (r arccos \sqrt{x/4})}{πx^{1−r/2}}.

For the special cases we use the identity sin (\frac{1}{2} arccos(t)) = \sqrt{(1 − t)/2}, which is valid for 0 ≤ t ≤ 1.

Remark. Note that

\frac{W_{2,1} (\sqrt{x})}{2\sqrt{x}} = \frac{1}{4}W_{2,1/2} (\frac{x}{4}) = \frac{\sqrt{4 − \sqrt{x}}}{4πx^{3/4}}.  (36)

It means that if X, Y are random variables such that X ∼ μ(2, 1) and Y ∼ μ(2, 1/2) then X^{2} ∼ 4Y. This can be also derived from the relation A_{m}(2, 1/2)4^{m} = A_{2m}(2, 1) = \binom{4n+1}{2n}/(4n + 1), A048990 in OEIS [24]. Hence A048990 is the moment sequence of the density function (36), x ∈ (0, 16).

4 Some particular cases

In this part we will see that for k = 3 some densities still can be represented as elementary functions. We need two families of formulas (cf. 15.4.17 in [19]).

Lemma 4.1. For c ≠ 0, −1, −2, … we have

2F_{1}( \frac{c}{2}, \frac{c − 1}{2}; c| z) = 2^{c−1}(1 + \sqrt{1 − z})^{1−c},  (37)
2F_{1}( \frac{c + 1}{2}, \frac{c − 2}{2}; c| z) = \frac{2^{c−1}}{c} (1 + \sqrt{1 − z})^{1−c}(c − 1 + \sqrt{1 − z}). (38)

Proof. We know that _{2}F_{1}(a, b; c| z) is the unique function f which is analytic at z = 0, with f(0) = 1, and satisfies the hypergeometric equation:

z(1 − z)f′′(z) + [c − (a + b + 1)z]f′(z) − abf(z) = 0

(see [3]). Now one can check that this equation is satisfied by the right hand sides of (37) and (38) for given parameters a, b, c.

Now consider p = 3/2. Theorem 4.2. Assume that p = 3/2. Then for r = 1/2, 1, 3/2 we have

W_{3/2,1/2}(x) = \frac{(1 + \sqrt{1 − 4x^{2}/27})^{2/3} − (1 − \sqrt{1 − 4x^{2}/27})^{2/3}}{2^{5/3}3^{−1/2}πx^{2/3}} ,  (39)
W_{3/2,1}(x) = 3^{1/2} \frac{(1 + \sqrt{1 − 4x^{2}/27})^{1/3} − (1 − \sqrt{1 − 4x^{2}/27})^{1/3}}{2^{4/3}πx^{1/3}}  (40)

+3^{1/2}x^{1/3} \frac{(1 + \sqrt{1 − 4x^{2}/27})^{2/3} − (1 − \sqrt{1 − 4x^{2}/27})^{2/3}}{2^{5/3}π}

and, finally, W_{3/2,3/2}(x) = x · W_{3/2,1}(x), with x ∈ (0, 3\sqrt{3}/2). Proof. For arbitrary r we have

W_{3/2,r}(x) = \frac{2^{1−2r/3} sin (2πr/3)}{3^{3/2−r}π}_{3}F_{2}( \frac{3 + 2r}{6} , \frac{r}{3}, \frac{−2r}{3} ; \frac{2}{3}, \frac{1}{3}| z)z^{r/3−1/2}
−\frac{2^{(4−2r)/3}r sin ((1 − 2r)π/3)}{3^{3/2−r}π}_{3}F_{2}( \frac{5 + 2r}{6} , \frac{1 + r}{3} , \frac{1 − 2r}{3} ; \frac{4}{3}, \frac{2}{3}| z)z^{(r+1)/3−1/2}
−\frac{r(1 + 2r) sin ((1 + 2r)π/3)}{2^{(1+2r)/3}3^{3/2−r}π}_{3}F_{2}( \frac{7 + 2r}{6} , \frac{2 + r}{3} , \frac{2 − 2r}{3} ; \frac{5}{3}, \frac{4}{3}| z)z^{(r+2)/3−1/2},

where z = 4x^{2}/27. If r = 1/2 or r = 1 then one term vanishes and in the two others the hypergeometric functions reduce to _{2}F_{1}. For r = 1/2 we apply (37) to obtain:

W_{3/2,1/2}(x) = \frac{z^{−1/3}}{2^{1/3}3^{1/2}π} _{2}F_{1}( \frac{1}{6}, \frac{−1}{3}; \frac{1}{3}| z) − \frac{z^{1/3}}{2^{5/3}3^{1/2}π} _{2}F_{1}( \frac{5}{6}, \frac{1}{3}; \frac{5}{3}| z)

= \frac{z^{−1/3}}{2^{1/3}3^{1/2}π} 2^{−2/3} (1 + \sqrt{1 − z})^{2/3} − \frac{z^{1/3}}{2^{5/3}3^{1/2}π} 2^{2/3} (1 + \sqrt{1 − z})^{−2/3}

= \frac{z^{−1/3}}{2 · 3^{1/2}π} (1 + \sqrt{1 − z})^{2/3} − \frac{z^{1/3}}{2 · 3^{1/2}π} (\frac{1 − \sqrt{1 − z}}{z} )^{2/3}

= \frac{z^{−1/3}}{2 · 3^{1/2}π} (1 + \sqrt{1 − z})^{2/3} − \frac{z^{−1/3}}{2 · 3^{1/2}π} (1 − \sqrt{1 − z})^{2/3}

and this yields (39). For r = 1 we use (38):

W_{3/2,1}(x) = \frac{z^{−1/6}}{2^{2/3}π} _{2}F_{1}( \frac{5}{6}, \frac{−2}{3}; \frac{2}{3}| z) + \frac{z^{1/6}}{2^{1/3}π} _{2}F_{1}( \frac{7}{6}, \frac{−1}{3}; \frac{4}{3}| z)

Documenta Mathematica 18 (2013) 1573–1596

= \frac{z^{−1/6}}{4π} (1 + \sqrt{1 − z})^{1/3}(3\sqrt{1 − z} − 1) + \frac{z^{1/6}}{4π} (1 + \sqrt{1 − z})^{−1/3}(3\sqrt{1 − z} + 1)

= \frac{z^{−1/6}}{4π} (1 + \sqrt{1 − z})^{1/3}(3\sqrt{1 − z} − 1)+ \frac{z^{−1/6}}{4π} (1 − \sqrt{1 − z})^{1/3}(3\sqrt{1 − z} + 1).

Now we have

(1 + \sqrt{1 − z})^{1/3} (3\sqrt{1 − z} − 1) = − (1 + \sqrt{1 − z})^{1/3} (3 − 3\sqrt{1 − z} − 2)
= −3z^{1/3} (1 − \sqrt{1 − z})^{2/3} + 2 (1 + \sqrt{1 − z})^{1/3}

and similarly

(1 − \sqrt{1 − z})^{1/3} (3\sqrt{1 − z} + 1) = 3z^{1/3} (1 + \sqrt{1 − z})^{2/3} − 2 (1 − \sqrt{1 − z})^{1/3} .

Therefore

W_{3/2,1}(x) = \frac{z^{−1/6}}{2π} ((1 + \sqrt{1 − z})^{1/3} − (1 − \sqrt{1 − z})^{1/3})

+\frac{3z^{1/6}}{4π} ((1 + \sqrt{1 − z})^{2/3} − (1 − \sqrt{1 − z})^{2/3}) ,

which entails (40). The last statement is a consequence of (22).

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, see (4.4) in [26]. The integer sequence

4^{m}A_{m}(3/2, 1/2) = \binom{3m/2 + 1/2}{n}\frac{4^{m}}{3m + 1},

moments of the density function W_{3/2,1/2}(x/4)/4 on the interval (0, 6\sqrt{3}), appears as A078531 in [24] and counts the number of symmetric noncrossing connected graphs on 2n + 1 equidistant nodes on a circle. The axis of symmetry is a diameter of a circle passing through a given node, see [7].

The measure μ(3/2, 1) is equal to μ(2, 1)^{⊠1/2}, the multiplicative free square root of the Marchenko-Pastur distribution and the integer sequence

4^{m}A_{m}(3/2, 1) = \binom{3m/2 + 1}{n}\frac{4^{m}}{3m/2 + 1},

moments of the density function W_{3/2,1}(x/4)/4 on (0, 6\sqrt{3}), appears in [24] as A214377.

For the sake of completeness we also include the densities for the sequences A_{m}(3, 1) (A001764 in [24]) and A_{m}(3, 2) (A006013), which have already appeared in [20, 21].

Densities of the Raney Distributions 1587

Theorem 4.3. Assume that p = 3. Then for r = 1, 2, 3 we have

W_{3,1}(x) = \frac{3 (1 + \sqrt{1 − 4x/27})^{2/3} − 2^{2/3}x^{1/3}}{2^{4/3}3^{1/2}πx^{2/3} (1 + \sqrt{1 − 4x/27})^{1/3}} ,  (41)
W_{3,2}(x) = \frac{9 (1 + \sqrt{1 − 4x/27})^{4/3} − 2^{4/3}x^{2/3}}{2^{5/3}3^{3/2}πx^{1/3} (1 + \sqrt{1 − 4x/27})^{2/3}}  (42)

and, finally, W_{3,3}(x) = x · W_{3,1}(x), with x ∈ (0, 27/4). Proof. For arbitrary r we have

W_{3,r}(x) = \frac{2^{(6−2r)/3} sin (πr/3)}{3^{3−r}π} _{3}F_{2}( \frac{r}{3}, \frac{3 − r}{6} , \frac{−r}{6} ; \frac{2}{3}, \frac{1}{3}| z)z^{(r−3)/3}
−\frac{2^{(4−2r)/3}r sin ((1 + r)π/3)}{3^{3−r}π} _{3}F_{2}( \frac{1 + r}{3} , \frac{5 − r}{6} , \frac{2 − r}{6} ; \frac{4}{3}, \frac{2}{3}| z)z^{(r−2)/3}
+\frac{r(r − 1) sin ((1 − r)π/3)}{2^{(1+2r)/3}3^{3−r}π} _{3}F_{2}( \frac{2 + r}{3} , \frac{7 − r}{6} , \frac{4 − r}{6} ; \frac{5}{3}, \frac{4}{3}| z)z^{(r−1)/3},

where z = 4x/27. For r = 1 and r = 2 we have similar reduction as in the previous proof. Here we will be using only (37). Taking r = 1 we get

W_{3,1}(x) = \frac{2^{1/3}z^{−2/3}}{3^{3/2}π} _{2}F_{1}( \frac{1}{3}, \frac{−1}{6}; \frac{2}{3}| z) − \frac{z^{−1/3}}{2^{1/3}3^{3/2}π} _{2}F_{1}( \frac{2}{3}, \frac{1}{6}; \frac{4}{3}| z)

= \frac{z^{−2/3}}{3^{3/2}π} (1 + \sqrt{1 − z})^{1/3} − \frac{z^{−1/3}}{3^{3/2}π} (1 + \sqrt{1 − z})^{−1/3}

= \frac{(1 + \sqrt{1 − z})^{2/3} − z^{1/3}}{3^{3/2}πz^{2/3} (1 + \sqrt{1 − z})^{1/3}} ,

which implies (41). Now we take r = 2:

W_{3,2}(x) = \frac{z^{−1/3}}{2^{1/3}3^{1/2}π} _{2}F_{1}( \frac{1}{6}, \frac{−1}{3}; \frac{1}{3}| z) − \frac{z^{1/3}}{2^{5/3}3^{1/2}π} _{2}F_{1}( \frac{5}{6}, \frac{1}{3}; \frac{5}{3}| z)

= \frac{z^{−1/3}}{2 · 3^{1/2}π} (1 + \sqrt{1 − z})^{2/3} − \frac{z^{1/3}}{2 · 3^{1/2}π} (1 + \sqrt{1 − z})^{−2/3}

= \frac{(1 + \sqrt{1 − z})^{4/3} − z^{2/3}}{2 · 3^{1/2}πz^{1/3} (1 + \sqrt{1 − z})^{2/3}} ,

and this gives us (42). Finally we apply (22). Recall that the measure μ(3, 1) is equal to μ(2, 1)^{⊠2}, the multiplicative free square of the Marchenko-Pastur distribution.

Documenta Mathematica 18 (2013) 1573–1596

Figure 1: Raney distributions W_{3/2,r}(x) with values of the parameter r labeling each curve. For r > p solutions drawn with dashed lines are not positive.

5 Graphical representation of selected cases

The explicit form of W_{p,r}(x) given in Theorem 3.2 permits a graphical visualization for any rational p > 0 and arbitrary r > 0. We shall represent some selected cases in Figures 1–9. These graphs which are partly negative are drawn as dashed curves. In Fig. 1 the graphs of the functions W_{3/2,r}(x) for values of r ranging from 0.9 to 2.3 are given. For r ≤ 3/2 these functions are positive, otherwise they develop a negative part. In Fig. 2 we represent W_{5/2,r}(x) for r ranging from 2 to 3.4. In Fig. 3 we display the densities W_{p,p}(x) for p = 6/5, 5/4, 4/3 and 3/2. All these densities are unimodal and vanish at the extremities of their supports. They can be therefore considered as generalizations of the Wigner’s semicircle distribution W_{2,2}(x), see equation (23). In Fig. 4 we depict the functions W_{4/3,r}(x), for values r ranging from 0.8 to 2.4. Here for r ≥ 1.4 negative contributions clearly appear. In Fig. 5 and 6 we present six densities expressible through elementary functions, namely W_{3/2,r}(x) for r = 1/2, 1, 3/2, see Theorem 4.2 and W_{3,r}(x) for r = 1, 2, 3, see Theorem 4.3. In Fig. 7 the set of densities W_{p,1}(x) for five fractional values of p is presented. The appearance of maximum near x = 1 corresponds to the fact that μ(p, 1) weakly converges to δ_{1} as p → 1^{+}. In Fig. 8 the fine details of densities W_{p,1}(x) for p = 5/2, 7/3, 9/4, 11/5, on a narrower range 2 ≤ x ≤ 4.5 are presented. In Fig. 9 we display the densities W_{p,1}(x) for p = 2, 5/2, 3, 7/2, 4, near the upper edge of their respective supports, for 3 ≤ x ≤ 9.5.

Figure 2: As in Fig. 1 for Raney distributions W_{5/2,r}(x).

Figure 3: Diagonal Raney distributions W_{p,p}(x) with values of the parameter p labeling each curve.

Figure 4: The functions W_{4/3,r}(x) for r ranging from 0.8 to 2.4.

Figure 5: Raney distributions W_{3/2,r}(x) with values of the parameter r labeling each curve. The case W_{3/2,1}(x) represents MP^{⊠1/2}, the multiplicative free square root of the Marchenko-Pastur distribution.

Figure 6: Raney distributions W_{3,r}(x) with values of the parameter r labeling each curve. The case W_{3,1}(x) represents MP^{⊠2}, the multiplicative free square of the Marchenko-Pastur distribution.

Figure 7: Raney distributions W_{p,1}(x) with values of the parameter p labeling each curve. The case W_{3/2,1}(x) represents the multiplicative free square root of the Marchenko–Pastur distribution, MP^{⊠1/2}.

Figure 8: Tails of the Raney distributions W_{p,1}(x) with values of the parameter p labeling each curve.

Figure 9: As in Fig. 8 for larger values of the parameter p.

Figure 10: Parameter plane (p, r) describing the Raney numbers. The shaded set Σ corresponds to nonnegative probability measures μ(p, r). The vertical line p = 2 and the stars represent values of parameters for which W_{p,r}(x) is an elementary function. Here MP denotes the Marchenko–Pastur distribution, MP^{⊠s} its s-th free mutiplicative power, B-the Bures distribution while SC denotes the semicircle law. For p > 1 the points (p, p) on the upper edge of Σ represent the generalizations of the Wigner semicircle law, see Fig. 3.

The Fig. 10 summarizes our results in the p > 0, r > 0 quadrant of the (p, r) plane, describing the Raney numbers (c.f. Fig. 5.1 in [16] and Fig. 7 in [21]). The shaded region Σ indicates the probability measures μ(p, r) (i.e. where W_{p,r}(x) is a nonegative function). The vertical line p = 2 and the stars indicate the pairs (p, r) for which W_{p,r}(x) is an elementary function, see Corollary 3.3, Theorem 4.2 and Theorem 4.3. The points (3/2, 1) and (3, 1) correspond to the multiplicative free powers MP^{⊠1/2} and MP^{⊠2} of the Marchenko-Pastur distribution MP. Symbol B at (3/2, 1/2) indicates the Bures distribution and SC at (2, 2) denotes the semicircle law centered at x = 2, with radius 2.

It is our pleasure to thank M. Bożejko, Z. Burda, K. Górska, I. Nechita and M. A. Nowak for fruitful interactions.

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W. Młotkowski Instytut Matematyczny, Uniwersytet Wrocławski Plac Grunwaldzki 2/4 50-384 Wrocław, Poland mlotkow@math.uni.wroc.pl K. A. Penson Laboratoire de Physique Théorique de la Matière Condensée (LPTMC) Université Pierre et Marie Curie CNRS UMR 7600 Tour 13 - 5ième ét. Boîte Courrier 121 4 place Jussieu F 75252 Paris Cedex 05 France penson@lptl.jussieu.fr

K. Życzkowski Institute of Physics Jagiellonian University ul. Reymonta 4 30-059 Kraków, Poland and Center for Theoretical Physics Polish Academy of Sciences al. Lotników 32/46 02-668 Warszawa, Poland karol@tatry.if.uj.edu.pl