Theory of Barnes Beta Distributions

A new family of probability distributions $\beta_{M, N},$ $M=0\cdots N,$ $N\in\mathbb{N}$ on the unit interval $(0, 1]$ is defined by the Mellin transform. The Mellin transform of $\beta_{M, N}$ is characterized in terms of products of ratios of Barnes multiple gamma functions, shown to satisfy a functional equation, and a Shintani-type infinite product factorization. The distribution $\log\beta_{M, N}$ is infinitely divisible. If $M<N,$ $-\log\beta_{M, N}$ is compound Poisson, if $M=N,$ $\log\beta_{M, N}$ is absolutely continuous. The integral moments of $\beta_{M, N}$ are expressed as Selberg-type products of multiple gamma functions. The asymptotic behavior of the Mellin transform is derived and used to prove an inequality involving multiple gamma functions and establish positivity of a class of alternating power series. For application, the Selberg integral is interpreted probabilistically as a transformation of $\beta_{1, 1}$ into a product of $\beta^{-1}_{2, 2}s.$

The main contribution of this paper is to construct and study the main properties of a novel family of probability distributions on the unit interval (0, 1] that naturally generalize the beta distribution to arbitrary multiple gamma functions.In particular, we show that all of these distributions have infinitely divisible logarithm, satisfy a functional equation and several symmetries, and admit a remarkable infinite-product factorization.We call them Barnes beta distributions.
Our paper contributes to several areas of current interest in probability theory.First, we contribute to the probabilistic study of Barnes multiple gamma functions and, more generally, the study of infinite divisibility in the context of special functions of analytic number theory complementing [3], [11], [15], [16].We show that a new class of Lévy-Khinchine representations is naturally associated with multiple gamma functions.Moreover, the meromorphic functions that we introduce as the Mellin transform of the Barnes beta distributions appear to have a number analytic significance as their pole structure depends on the rationality of the parameters of the distribution.
Second, there is a long-standing interest in the literature in the study of Dufresne distributions, whose defining property is that their Mellin transform is given in the form of a product of ratios of Euler's gamma functions, confer [5], [6], and references therein.In addition, there have recently appeared a series of papers [10], [13], [14] that computed the Mellin transform of a certain functional of the stable process in the form of a product of ratios of Alexeiewsky-Barnes G−functions (or, equivalently, the double gamma function).In our own work on the Selberg integral [18] we introduced a different probability distribution having the same property that its Mellin transform is given in the form of a finite product of ratios of G−factors.The contribution of this paper is to show that there is a whole family of Barnes beta distributions on the unit interval (0, 1] that extends this property to arbitrary multiple gamma functions.
Third, we contribute to the probabilistic theory of the Selberg integral complementing [24].We show that the Selberg integral extends as a function of its dimension to the Mellin transform of a probability distribution, which factorizes in terms of β −1 2,2 s.This leads us to a new interpretation of the Selberg integral and extends the results of [18].Fourth, motivated by [3], we introduce a new family of infinitely divisible distributions that is related to the Riemann ξ function and express it in terms of a limit of Barnes beta distributions by means of a functional equation for the Mellin transform, thereby advancing the probabilistic theory of ξ.
As an application of our results, we introduce a novel class of power series, compute their Mellin transform, and prove their positivity by relating them to the Laplace transform of the Barnes beta distribution.
The main technical tool that we rely on in this paper is the remarkable approach to multiple gamma functions due to Ruijsenaars [19].Ruijsenaars developed a novel Malmstén-type formula for a class of functions that includes the multiple log-gamma function as a special case.We prove the key infinitely divisibility property in complete generality, that is for the whole Ruijsenaars class, before specializing to the gamma functions.Most of our proofs are elementary as the strength of his approach allows us to reduce our arguments to simple properties of multiple Bernoulli polynomials.
The plan of the paper is as follows.In Section 1 we remind the reader of the basic properties of the Barnes gamma functions following [2] and [19].In Section 2 we state our main results.Section 3 gives examples of Barnes beta distributions.Section 4 explains the connection between β 1,1 , β 2,2 , and the Selberg integral.In Section 5 we introduce a limit of Barnes beta distributions that approximates the Riemann ξ function.
In Section 6 we present the proofs.Section 7 concludes with a summary.

Review of Multiple Gamma Functions
Let f (t) be of the Ruijsenaars class, i.e. analytic for (t) > 0 and at t = 0 and of at worst polynomial growth as t → ∞, confer [19], Section 2. The main example that corresponds to the case of Barnes multiple gamma functions is for some integer M ≥ 0 and parameters a j > 0, j = 1 • • • M. For concreteness, the reader can assume with little loss of generality that f (t) is defined by (1.1).Slightly modifying the definition in [19], we define generalized Bernoulli polynomials by (1. 2) The generalized zeta function is defined by It is shown in [19] that ζ M (s, w) has an analytic continuation to a function that is meromorphic in s ∈ C with simple poles at s = 1, 2, • • • M. The generalized log-gamma function is then defined by (1.4) It can be analytically continued to a function that is holomorphic over C − (−∞, 0].The key results of [19] that we need are summarized in the following theorem. Theorem 1.1 (Ruijsenaars).L M (w) satisfies the Malmstén-type formula for (w) > 0, M (w) . (1.5) L M (w) satisfies the asymptotic expansion, (1.7) In the special case of the function f (t) being defined by (1.1), the generalized zeta and gamma functions have important additional properties.It is not difficult to show that (1.3) becomes for a = (a 1 , • • • , a M ), which is the formula given originally by Barnes [2] for the multiple zeta function.Let L M (w | a) be defined by (1.4) with f (t) as in (1.1).Now, following [19], define3 the Barnes multiple gamma function by (1.9) It follows from (1.8) and (1.9) that Γ M (w | a) satisfies the fundamental functional equation , and Γ 0 (w) = 1/w, which is also due to [2].By iterating (1.10) one sees that Γ M (w | a) is meromorphic over C having no zeroes and poles at with multiplicity equal the number of M −tuples (k 1 , • • • , k M ) that satisfy (1.11).
We conclude our review of the Barnes functions by relating the general results to the classical case of Euler's gamma and Hurwitz's zeta functions.Following [19], we have the identities ) and the asymptotic expansion in Theorem 1.1 becomes Stirling's series.

Barnes Beta Distribution
In this section we will define and describe the main properties of what we call Barnes beta distributions.We begin by introducing a combinatorial operator S N that plays a central role in the formulation of our results.
Let {b k }, k ∈ N be a sequence of positive real numbers and N, M ∈ N. Define the action of the operator S N by Definition 2.1.
In other words, in (2.1) the action of S N is defined as an alternating sum over all combinations of p elements for every p = 0 • • • N. Given a function f (t) of Ruijsenaars class, confer Section 1, such that f (t) > 0 for t ≥ 0, let L M (w) be the corresponding generalized log-gamma function defined in (1.4).The main example is the function f (t) in (1.1) so that L M (w) = L M (w | a) is the Barnes multiple log-gamma function.We can now define the main object that we will study in this paper.
The function η M,N (q | b) is holomorphic over q ∈ C − (−∞, −b 0 ] and equals a product of ratios of generalized gamma functions by construction.Denoting Γ M (w) = exp L M (w) , it is easy to write out examples of η M,N (q | b) for small N.
We now proceed to state our results. 4We begin with the general case and then specialize to that of multiple gamma functions.

Corollary 2.15 (Solution to Functional Equations).
The infinite product representation in Theorem 2.13 is the solution to the functional equation in Theorem 2.8.

Corollary 2.16 (Barnes Beta Algebra
(2.30) We conclude this section with two results that hold for special values of a and b.
Theorem 2.17 (Reduction to Independent Factors).Given i, j, if b j = n a i for n ∈ N, (2.31) We note that the structure of β M,N (a) depends on rationality of (a . This is clear from Definition 2.2 as this structure is determined by ratios of multiple gamma functions that have poles specified in (1.11).This phenomenon was studied in a different context for the double gamma function in [10] and [13].

Examples
It is not difficult to compute β M,N (a, b) for small M and N. We give four examples that can be checked by direct inspection.
In the rest of this section we will focus on the special case of M = N = 2 in order to illustrate the general theory with a concrete yet quite non-trivial example.In addition, this case is also of a particular interest in the probabilistic theory of the Selberg integral that we will review in Section 4. Let a 1 = 1 and a 2 = τ > 0 and write Using (1.12), the functional equation in Theorem 2.8 takes the form The positive moments in Corollary 2.10 for k ∈ N are The negative moments are The positivity conditions in Corollary 2.11 for x > 0 are Corollary 2.12 gives for 0 < (q) < b 0 and 0 < (q) < b 0 /τ, respectively, The factorization equations in Theorem 2.13 are Finally, the positive integral moments in case of τ = 1 take on a generalized hypergeometric form by Theorem 2.18.Let (b The density of β 2,2 (τ, b) can be computed by Laplace transform inversion.This computation requires a separate study similar to [10] as the structure of the residues of η 2,2 (q | τ, b) depends on the rationality of τ, b 1 , and b 2 .

β 2,2 (τ, b) and Selberg Integral
In this section we will review the application of β 2,2 (τ, b) to the probabilistic structure of the celebrated Selberg integral that we developed in [18] using special properties of the Alexeiewsky-Barnes G−function.The goal of re-formulating this structure here is to put it into the general framework of the Barnes beta distributions, which gives it a novel derivation and leads to a new interpretation of the Selberg integral.
The starting point of the probabilistic study of the Selberg integral is the following remarkable formula due to Selberg [20].Given 0 < µ < 2, λ i > −µ/2, and 1 ≤ l < 2/µ, The reader who is familiar with the classical approach to the Selberg integral, confer [8], will notice that we have written Selberg's formula in a somewhat peculiar form.The reason for restricting 0 < µ < 2 is that in this case, given a general function ϕ(s), S µ,l [ϕ] equals the lth moment of the probability distribution constructed by integrating ϕ(s) with respect to the limit lognormal stochastic measure, 5 confer [18].In an attempt to compute this distribution for ϕ(s) = s λ1 (1 − s) λ2 , in [18] we constructed 6 and factorized a probability distribution 7 having the moments given by Selberg's formula in (4.1). 5 We will not attempt to quantify this statement here as it would take us too far afield and it is solely used to motivate Theorem 4.1 and our interpretation of the Selberg integral in Remark 4.4. 6In the special case of λ 1 = λ 2 = 0 an equivalent formula for the Mellin transform first appeared in [17].
The general case was first considered by [9], who gave an equivalent expression for the right-hand side of (4.3) and so matched the moments without proving that it corresponds to a probability distribution.
We can now relate the Selberg integral and β 2,2 (τ, b).Let τ > 1 and define i.e. log L is a zero-mean normal with variance 4 log 2/τ and Y is a power of the exponential.Given  where L is as in (4.6) and N is some distribution that is independent of L. If the negative moments of M (µ,λ1,λ2) equal those of M (µ,λ1,λ2) in (4.5), then M (µ,λ1,λ2) in law = M (µ,λ1,λ2) .
i.e. the Selberg integral can be interpreted probabilistically as a transformation of β 1,1 into the product in (4.10).
It is an open question how to extend this mechanism to β M,M , i.e. how to compute a probability distributions having S µ,l pdf of β M,M as its moments.
We conclude this section with a "master theorem" for arbitrary a 1 , a 2 > 0, which implies Theorems 4.1 and 4.2.The proofs are given in Section 6.

An Approximation of the Riemann ξ function
We remind the reader of the definition of the Riemann ξ function following [3], which is the primary reference as well as motivation for the results of this section.
ξ(q) 1 2 q(q − 1)π −q/2 Γ(q/2)ζ(q). (5.1) The function ξ(q) has a remarkable probabilistic representation, originally due to Riemann, which in the modern language 9 can be written in the form where S 2 is a probability distribution on (0, ∞) that is defined by and {Γ 2,n } denotes an iid family of gamma distributions on (0, ∞) with the density xe −x .
Recalling the Jacobi theta function θ(t) (strictly speaking, a special case of Jacobi's θ 3 ) the Laplace transform of S 2 can be written as ) − 1 dt t , q > 0. (5.6) In particular, S 2 is infinitely divisible and absolutely continuous. 10his construction inspired us to try to find a probabilistic representation of ξ in terms of the Mellin transform of Barnes beta distributions.The relevance of the latter is that the Jacobi triple product for θ(t) naturally leads to a limit of Barnes beta distributions.
Proof of Theorem 2.4.Let M ≤ N and (q) > −b 0 .We start with Definition 2.2 and substitute (1.5) for L M (w).By (6.8) in Corollary 6.2 and linearity of S N , we obtain (6.12) Letting r = 0 in (6.7), we have the identity so that (6.12) can be simplified to This is the canonical representation of the Laplace transform of an infinitely divisible distribution on [0, ∞), confer Theorem 4.3 in Chapter 3 of [22].
Proof of Corollary 2.5.We note that  The result follows from (6.16).
Proof of Theorem 2.6.The starting point of the proof is (1.6).Substituting (1.6) into (2.2) and using linearity of S N , we can write in the limit of q → ∞, | arg(q Now, to compute S N B M (w) log(w) (q | b), we expand the logarithm in powers of 1/q, resulting in terms of the form (6.3) with n = M.By (6.2) in Lemma 6.1, if r + m > M, then such terms are of order O(1/q).If r + m ≤ M and M < N, they are all zero by (6.2).If r + m ≤ M and M = N, the only non-zero terms satisfy r + m = N so that they have degree zero in q.Hence, we have the estimate If M < N, the expression in (6.22) is zero by (6.8) and the sum in (6.21) is zero by (6.10) so that (2.8) follows from (6.21).If M = N, the result follows from (6.9) and (6.11).
Proof of Theorem 2.8.It is sufficient to substitute (1.10), written in the form into (2.2) and recall the definition of η M −1,N (q | âi , b).
Proof of Corollary 2.11.The absolute convergence of the series in (2.19) follows from Theorem 2.6 and (2.17).Its equality to the Laplace transform is the general property of the power series of positive integral moments, confer Section 7.6 of [7].
Proof of Corollary 2.12.The proof is a direct corollary of Ramanujan's Master Theorem, confer [1].It is only sufficient to note that η M,N (q | a, b) is analytic over (q) > −b 0 and, by Theorem 2.6, satisfies Hardy's growth conditions there.
Proof of Theorem 2. Proof of Corollary 2.15.By the infinite product representation in (2.24), we have by the induction assumption and (2.12).
Proof of Theorem 2.18.In the case of a i = 1 we can write (2.17) in the form By repeated application of (6.35) and the identity we obtain by induction on k = 0 The result follows by letting k = M − 1 and recalling that L 0 (w) = − log(w).
We now proceed to the proof of Theorem 4.5.The starting point is the following multiplication formula, originally due to Barnes [2] in a slightly different form.Our version can be seen as an elementary corollary of (1.5).Recall the definition of multiple Bernoulli polynomials corresponding to f (t) in Eq. (1.1), (1 − e −aj t ) −1 e −xt .
(6.38) Theorem 6.4 (Barnes multiplication).Let (z) > 0 and k = 1, 2, 3, The multiplication formula gives us Hence, we can write for E X q 1 E X q 2 E X q (6.43) Now, using the identity and the definition of L in Theorem 4.5, we can write for E L The proof of (4.4) and (4.5) is now immediate from (1.10), which implies the identity where Γ 1 (x | τ ) is defined in (1.12).The remaining computation, which determines the overall constant in (4.10), is straightforward and will be omitted.
Then, the Laplace transform of S 2 (δ) can be written as where we used Frullani's formula for log(x) to evaluate the integral in (6.57) and the infinite product representation of sinh(x) and (5.5) to obtain (6.59).This proves (6.51) and (6.52).The density of S 2 (δ) follows from (6.51) so that the Mellin transform is x q e −δx f S2 (x) dx. (6.60) Expanding the exponential and making use of (5.2), we obtain (6.54), provided that the integral can be computed term by term.The partial sums of exp(−δx) are bounded by exp(δx).If δ < π 2 /2, then exp(δx)f S2 (x) is exponentially small as x → ∞, confer Table 1 in [3] so that the result follows by dominated convergence.The series is absolutely convergent if δ < π 2 /2 as is clear from (5.1) since ζ(q) → 1 as q → ∞ and (q) > 0.
The main areas of applications that we considered in this paper are probabilistic theories of the Selberg integral and of Riemann ξ function.We constructed a distribution with the property that its positive moments are given by Selberg's formula, characterized its negative moments, and decomposed it into a product of β −1 2,2 (a, b)s.We showed

Theorem 2 . 4 (
Existence).Given M, N ∈ N such that M ≤ N, the function η M,N (q | b) is the Mellin transform of a probability distribution on (0, 1].Denote it by β M,N (b).Then,

.32) Remark 2 . 19 .
The generating function of the moments, i.e. the Laplace transform, in the case of a i = 1 for all i = 1 • • • M gives an interesting generalization of the confluent hypergeometric function corresponding to the Laplace transform for M = N = 1.

. 16 )
It remains to note that dK (f ) M,N (t | b) satisfies the required integrability condition

( 6 .
45)This proves that the expression on the right-hand side of (4.13) is in fact the Mellin transform of a probability distribution on (0, ∞).Its properties follow from the known properties of the normal and β 2,2 (a, b) distributions.Proof of Theorems 4.1, 4.2, and Corollary 4.3.Define Theorem 4.5, the difference between (4.3) and (4.13) is in the third gamma ratio, which accounts for the appearance of the Y distribution in(4.10).By the functional equation of Γ 2 in (1.10) and the definition of Y in (4.6), we have We constructed and studied the main properties of a novel class of what we called Barnes beta probability distributions β M,N (a, b).β M,N (a, b) is a distribution on (0, 1] that is parameterized by two sets of positive real numbers a = (a 1 , • • • , a M ) and b = (b 0 , • • • , b N ) and defined by its Mellin transform η M,N (q | a, b).We gave four different representations of η M,N (q | a, b).The defining representation is in the form of a product of ratios of Barnes multiple gamma functions Γ M (w | a), thereby generalizing the classic beta distribution.We used Malmstén-type formula of Ruijsenaars for log Γ M (w | a) to show that − log β M,N (a, b) is infinitely divisible on [0, ∞) by deriving its Lévy-Khinchine form and thus giving the 2nd representation of η M,N (q | a, b).The Lévy-Khinchine form allowed us to show that − log β M,N (a, b) is compound Poisson if M < N and absolutely continuous if M = N.We used the functional equation of Γ M (w | a) to derive a functional equation for η M,N (q | a, b) and thus to compute the integral moments of β M,N (a, b) in the form of Selberg-type products of Γ M −1 (w | a).The Ruijsenaars form of the asymptotic expansion of log Γ M (w | a) in the limit w → ∞ gave us the asymptotic of η M,N (q | a, b) in the limit q → ∞.We used this asymptotic in the case of M < N to give a probabilistic proof of an inequality involving multiple gamma functions.We also used it to show the convergence of the power series of moments of β M,N (a, b) to the Laplace transform.The resulting series of Selberg-type products of Γ M −1 (w | a) is therefore positive, giving an interesting application of the general theory.We related this series to η M,N (q | a, b) by Ramanujan's Master Theorem, giving the 3rd representation of η M,N (q | a, b).We solved the functional equation of η M,N (q | a, b) in the form of Shintani-type infinite products of η M −1,N (q | a, b) and η M −1,N −1 (q | a, b), resulting in the 4th representation.Finally, we established several symmetries of the Mellin transform in the form of functional equationsrelating η M,N (q | a, b) to η M,N −1 (q | a, b), η M −1,N (q | a, b), and η M −1,N −1 (q | a, b).We illustrated our theory of Barnes beta distributions with several examples.First, we considered two special cases of a and b.If b N is an integer multiple of a M , β M,N (a, b) decomposes into a product of β M −1,N −1 (a, b)s.If a i = 1 for all i, the moments of β M,N (a, b) are given by a multiple product generalizing the moments of the classic beta distribution.Second, in some elementary cases of small M and N we computed the density and weight at 1 of β M,N (a, b) exactly.Our main non-elementary example is β 2,2 (a, b), bj .This is done by induction on n.If n = 1, then the result follows from (2.6).Assume (6.33) holds for n − 1, i.e. b j = (n − 1) a i .By (2.15), we have [4] proof of Corollary 4.3 follows from Theorem 4.2 due to the determinacy of the Stieltjes moment problems for β 2,2 (compactly supported) and Y −1 (Carleman's criterion) and its indeterminacy for L, L −1 (lognormal) and β −1 2,2 , Y (infinite moments), confer[4], Sections 2.2 and 2.3.The proof of Theorem 5.1 requires an auxiliary result that we state and prove first.We need to define a one-parameter family of infinitely divisible distributions S 2 (δ) on (0, ∞) extending S 2 = S 2 (0).Let f S2 (x) denote the probability density of S 2 .S 2 (δ) is infinitely divisible and absolutely continuous.Denote its density by f S2(δ) (x).Then, its Laplace transform, density, and Mellin transform satisfy [3](6.54)Proof.The starting point is the formula given in[3], Section 3.2, for the Lévy density ρ(t) of the weighted sum of positive, independent, infinitely divisible distributions of the form n c n X n , where c n > 0 and X n has Lévy density ρ(t) for all n.
(6.55)The Lévy density of Γ 2,n is 2e −t /t so that the Lévy density