Selberg Integral as a Meromorphic Function

D. Ostrovsky (2013) “Selberg Integral as a Meromorphic Function,” International Mathematics Research Notices, Vol. 2013, No. 17, pp. 3988–4028 Advance Access Publication July 13, 2012 doi:10.1093/imrn/rns170 Selberg Integral as a Meromorphic Function Dmitry Ostrovsky1,2 1Department of Mathematics, Yale University, New Haven, CT 06520, USA and 2125 Field Point Road, No. 3, Greenwich, CT 06830, USA Correspondence to be sent to: dm [email protected] The Selberg integral (the integral of the discriminant on n variables raised to a power −μ/2 times the beta prefactor ∏i xλ1i (1− xi)λ2 over the unit n-interval) is analytically extended as a function of its dimension n to the complex plane. The resulting meromor- phic functionM(q | μ, λ1, λ2) is expressed in terms of the Alexeiewsky–Barnes G-function and shown to satisfy two functional equations. Its values for negative integral q are computed in the form of a novel Selberg-type product of gamma factors. For the range of powers 0< μ < 2, this function is the Mellin transform of an absolutely continuous probability distribution on the positive real line having infinitely divisible logarithm. The small μ asymptotic expansion ofM(q | μ, λ1, λ2) is shown to coincide with the inter- mittency expansion of the Mellin transform of a particular functional of the limit log- normal stochastic process resulting in the conjecture that M(q | μ, λ1, λ2) is its Mellin transform. For application, we introduce two novel power series, compute their Mellin transforms, and prove their positivity by relating them to the structure of the underlying probability distribution. 1 Introduction The Selberg integral was introduced by Atle Selberg in 1944 (cf. [43]). It is a fascinating mathematical object that continues to generate substantial interest due to its ubiquitous Received November 29, 2011; Revised May 20, 2012; Accepted June 12, 2012 Communicated by Prof. Anton Alekseev c© The Author(s) 2012. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3989 appearance in statistical mechanics [18] and random matrix theory [34], occurrence in special function theory [4] and representation theory [48], and conjectured relevance to number theory [26]. The continued fascination with the Selberg integral is also due to the astounding originality of Selberg’s derivation coupled with the notorious difficulty of computing generalizations. We refer the reader to [19] for a comprehensive review. The starting point of this paper is the recently discovered connection of the Selberg integral to the so-called limit lognormal stochastic process. This process was conceived by Benoit Mandelbrot in 1972 (cf. [31]), reviewed by him in [32], and formal- ized in a series of papers by Jean-Pierre Kahane (cf. [23–25]). In his papers, Kahane constructed a comprehensive theory of multiplicative chaos, which can be thought of as the theory of convergence of a particular class of positive martingales to cer- tain limit random measures. The limit lognormal process is the simplest nontrivial example of such a limit measure. This process lay largely dormant until it was redis- covered by Bacry et al. around 2000 and investigated and extended in a series of papers (cf. [6–8, 35]). Independently, Barral and Mandelbrot investigated in [12] the limit log-compound Poisson process as another example of multiplicative chaos and proved a number of key estimates that were used in [8] to prove many important properties of the limit lognormal process. The key feature of the limit lognormal pro- cess is that it is defined as a certain limit of the exponential functional of a two- dimensional, logarithmically correlated gaussian free field. Such constructions have recently attracted substantial interest as they occur in a wide spectrum of problems in mathematical physics ranging from conformal field theory [13, 41] and quantum gravity [16], to random energy models [20], to conformally invariant random curves in the plane [5]. We refer the reader to [2, 42] for state-of-the-art reviews and novel extensions. In this paper, we will focus on one key property of the limit lognormal distri- bution that was discovered in [7]: the positive integral moments of the limit probability distribution are given by the restricted Selberg integral of the same dimension as the order of the moment. (The terms “probability distribution” and “random variable” are used interchangeably in this paper.) This property is easily extended to the full Sel- berg integral as follows. Given a real number μ (the intermittency parameter) in the range 0< μ < 2, let dMμ(s) denote the limit lognormal random measure on the inter- val s ∈ [0,1] having the decorrelation length of 1. (We mention in passing that 0< μ < 2 is required for the nondegeneracy of ∫1 0 dMμ(s) and q< 2/μ for the finiteness of posi- tive moments as shown in [8]. We refer the reader to [39, 40] for reviews, applications to functionals other than the Mellin transform, and various extensions.) Then, for any at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 3990 D. Ostrovsky constants λ1, λ2 > −μ/2 the positive integral moments of the positive random variable∫1 0 s λ1(1− s)λ2 dMμ(s) satisfy for 1≤ l < 2/μ E [(∫1 0 sλ1(1− s)λ2 dMμ(s) )l] = ∫ [0,1]l l∏ i=1 sλ1i (1− si)λ2 l∏ i< j |si − sj|−μ ds1 · · ·dsl = l−1∏ k=0 Γ (1− (k+ 1)μ/2)Γ (1+ λ1 − kμ/2)Γ (1+ λ2 − kμ/2) Γ (1− μ/2)Γ (2+ λ1 + λ2 − (l + k− 1)μ/2) . (1) The original result of [7] corresponds to λ1 = λ2 = 0. The proof of Equation (1) is elemen- tary and is given in Section 4. It follows that the Mellin transform of ∫1 0 s λ1(1− s)λ2 dMμ(s) defined by E[( ∫1 0 s λ1(1− s)λ2 dMμ(s))q], q ∈ C, provides a natural analytic continuation of the Selberg integral as a function of the positive integral dimension l to the com- plex plane. (It is more natural to define the Mellin transform as ∫∞ 0 x q f(x)dx as opposed to the usual ∫∞ 0 x q−1 f(x)dx for our purposes.) Denote the Mellin transform by M(q | μ, λ1, λ2). This continuation then has the obvious property that for purely imagi- nary arguments it is the Fourier transform (characteristic function) of the logarithm of∫1 0 s λ1(1− s)λ2 dMμ(s). M(iq | μ, λ1, λ2) =E [ exp ( iq log ∫1 0 sλ1(1− s)λ2 dMμ(s) )] , q ∈ R. (2) In addition, it turns out that it is possible to derive the intermittency expansion of the Mellin transform in the limit μ → 0 of small intermittency. We show in Section 4 that E [(∫1 0 sλ1(1− s)λ2 dMμ(s) )q] ∼ ( Γ (1+ λ1)Γ (1+ λ2) Γ (2+ λ1 + λ2) )q exp ( ∞∑ r=0 (μ 2 )r+1 1 r + 1 × [ (ζ(r + 1,1+ λ1) + ζ(r + 1,1+ λ2)) × ( Br+2(q) − Br+2 r + 2 ) − ζ(r + 1)q + ζ(r + 1) × ( Br+2(q + 1) − Br+2 r + 2 ) − ζ(r + 1,2+ λ1 + λ2) × ( Br+2(2q − 1) − Br+2(q − 1) r + 2 )]) . (3) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3991 As usual, Bn(s) denotes the nth Bernoulli polynomial, ζ(s,a) the Hurwitz zeta function, and ζ(1,a)�−ψ(a) the digamma function. Our main contribution is the construction of an explicit meromorphic function that satisfies Equations (1)–(3) in Theorem 2.4, which for the purpose of this paper are taken as axioms, discovery that the underlying probability distribution has infinitely divisible logarithm in Theorem 2.8, and computation of its fine probabilistic struc- ture in Theorem 3.7. Thus, M(q | μ, λ1, λ2) recovers Selberg’s formula for positive inte- gral q< 2/μ, is the Mellin transform of a probability distribution for �(q) < 2/μ as long as 0< μ < 2, and has the asymptotic expansion in the limit μ → 0 that is given in Equation (3). This advances the existing literature on the Selberg integral in the fol- lowing ways. The first paper that considered the problem of analytically extending the Selberg integral (with negative exponent, see below) as a function of its dimension was [38]. There we found and gave several equivalent representations of a function that is analytic in the half-plane �(q) < 2/μ and satisfies Equations (1)–(3) for the restricted Selberg integral corresponding to λ1 = λ2 = 0. Subsequently, Fyodorov et al. [20] inde- pendently found a solution for the full integral that satisfies Equation (1) only and stated that it is the same as ours in the restricted case. They formulated their solution in terms of a special function that was first introduced by Fateev et al. [17] and satis- fies functional equations, which resemble those of the Alexeiewsky–Barnes G-function. (The Alexeiewsky–Barnes G(z | τ) function is closely related to the Barnes double gamma function Γ2(z | ω1, ω2) and is not to be confused with the Barnes G(z) function. The two are related by G(z) =G(z | 1). See the appendix for precise definitions.) Our contribution is then to unify the two approaches by giving a representation ofM(q | μ, λ1, λ2) in terms of the Alexeiewsky–Barnes G-function. We show that the framework of the Alexeiewsky– Barnes G-function is the correct framework for the Selberg integral as the properties of the Alexeiewsky–Barnes G-function naturally imply those of M(q | μ, λ1, λ2). As an application of our results, we introduce two novel power series, compute their Mellin transforms, and interpret them probabilistically thereby proving their positivity. We conjecture thatM(q | μ, λ1, λ2) is the Mellin transform of ∫1 0 s λ1(1− s)λ2 dMμ(s). The function M(q | μ, λ1, λ2) is represented as an infinite product of ratios of gamma factors and, equivalently, as a finite product of ratios of G-factors. The infinite product extends Selberg’s finite product, whereas the product of G-factors gives the desired asymptotic expansion and probabilistic interpretation as the Mellin transform. Our analysis of the fine structure ofM(q | μ, λ1, λ2) leads us to introduce in Corollary 3.6 a novel class of positive probability distributions having the property that its Mellin transform is of the form of a finite product of ratios of G-factors. We note that a different at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 3992 D. Ostrovsky probability distribution having this property has recently been computed in [28], which leads to the interesting problem of classifying all such distributions. It is worth clarifying how our results advance and differ from the known results on the Selberg integral that have been established in the random matrix theory (cf. [9, 26]). The main difference is that the Selberg integrand exponent or at least its real part is negative in our case, whereas in the random matrix theory it is interpreted as inverse temperature and must be positive. This difference is essential as the key inter- pretation of our analytic continuation of the Selberg integral as the Mellin transform of a probability distribution holds only for a range of negative exponents. The main technical advance of our results is the use of the Alexeiewsky–Barnes G-function as opposed to the Barnes G-function and of Shintani’s identity in the context of the Selberg integral. The primary focus of our study in this paper is the Selberg integral, however, we also contribute to the theory of the limit lognormal stochastic process by giving a derivation of the intermittency expansion in Equation (3) and thereby providing motiva- tion for our main results. In [36, 37], we established a novel invariance of the underlying gaussian field with respect to the intermittency parameter and showed how it trans- lates into an equation for the intermittency derivative of a class of functionals of the limit process. We showed that such functionals admit a formal power series expansion with certain universal coefficients, which are determined uniquely by the restricted Sel- berg integral, and applied it in [38] to derive the expansion for the Mellin transform of∫1 0 dMμ(s). The contribution of this paper is to extend this technique to a more general class of functionals resulting in Equation (3). An additional contribution of this paper is to give a review of some of the prop- erties of Alexeiewsky–Barnes G-function that play a key role in our calculations. This special function was introduced by Alexeiewsky [1] and developed by Barnes [10], who established its main properties. We give explicit formulas for what Barnes called dou- ble gamma modular functions, a new derivation of the Malmste´n-type formula for the logarithm of the Alexeiewsky–Barnes G-function originally due to Lawrie and King [29], and a new proof of Shintani’s identity [44]. Our contribution is to show that all of these results are elementary corollaries of the fundamental characterization of the double gamma modular functions given by Barnes [10]. Our results are mathematically rigorous except in Section 4, where they are alge- braically exact at the level of formal power series expansions. The plan of our paper is as follows. In Sections 2 and 3, we state our results. In Section 4, we give a derivation of the expansion in Equation (3). In Section 5, we give the at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3993 proofs. Section 6 presents conclusions and some open questions. In the appendix, we review the Alexeiewsky–Barnes G-function. 2 Selberg Integral as a Meromorphic Function In this section, we will define a function of the variables q ∈ C, μ ∈ C, and λ1, λ2 ∈ R having the properties that were introduced in Equations (1)–(3). It is more natural to use the variable 2/μ instead of μ so that from now on we let τ � 2 μ , 0< � ( 1 τ ) < 1, λi > −� ( 1 τ ) . (4) Obviously, 0< �(1/τ) < 1 is equivalent to 0< �(μ) < 2 and �(λiτ) > −�(τ )�(1/τ) > −1. (We are mainly interested in 0< μ < 2 as explained in the Introduction so that the reader can let τ > 1 without loss of generality.) Definition 2.1. Given �(q) < �(τ ), define M(q | μ, λ1, λ2)� τ qΓ (1− q/τ)Γ (2− 2q + τ(1+ λ1 + λ2)) Γ q(1− 1/τ)Γ (2− q + τ(1+ λ1 + λ2)) ∞∏ m=1 (mτ)2q Γ (1− q +mτ) Γ (1+mτ) ×Γ (1− q + τλ1 +mτ) Γ (1+ τλ1 +mτ) Γ (1− q + τλ2 +mτ) Γ (1+ τλ2 +mτ) Γ (2− q + τ(λ1 + λ2) +mτ) Γ (2− 2q + τ(λ1 + λ2) +mτ) . (5) � It is easy to see that the infinite product is absolutely convergent.M(q | μ, λ1, λ2) has the following properties that will be proven in Section 5. (We note that in the special case of λ1 = λ2 = 0 Equation (5) and Theorems 2.2 and 2.6–2.8 first appeared in [38] under the simplifying assumptions that 0< μ < 1 and �(q) < 2/μ. A precursor of Theorem 2.4 with a different definition of G(z | τ) first appeared in [20]. The link between the infinite product in Equation (5) and the product in Theorem 2.4 and the results of Section 3 are new in all cases.) Theorem 2.2. The functionM(q | μ, λ1, λ2) satisfies for integral l = 1,2,3, . . . M(l | μ, λ1, λ2) = l−1∏ k=0 Γ (1− (k+ 1)/τ)Γ (1+ λ1 − k/τ)Γ (1+ λ2 − k/τ) Γ (1− 1/τ)Γ (2+ λ1 + λ2 − (l + k− 1)/τ) , (6) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 3994 D. Ostrovsky M(−l | μ, λ1, λ2) = l−1∏ k=0 Γ (2+ λ1 + λ2 + (l + 2+ k)/τ)Γ (1− 1/τ) Γ (1+ λ1 + (k+ 1)/τ)Γ (1+ λ2 + (k+ 1)/τ)Γ (1+ k/τ) . (7) � Remark 2.3. The function M(q | μ, λ1, λ2) in Equation (5) is defined for all q ∈ C as will become clear from Theorem 2.4 and its proof. It is understood that both sides of Equation (6) can simultaneously vanish or be infinite if τ is real and l ≥ τ . � Recall the definition of the Alexeiewsky–Barnes G-function. Following Barnes [10], we denote it by G(z | τ). Let z∈ C, τ ∈ C such that |arg(τ )| < π, and Ω �mτ + n. G(z | τ)� exp ( a(τ ) z τ + b(τ ) z 2 2τ 2 ) z τ ∞∏ m,n=0 ′ ( 1+ z Ω ) exp ( − z Ω + z 2 2Ω2 ) , (8) where the prime indicates that the double product is over all nonnegative integers m and nexceptm=n= 0. The functions a(τ ) and b(τ ) are chosen in such a way that G(z | τ) satisfies the following normalization and functional equations: G(z= 1 | τ) = 1, (9) G(z+ 1 | τ) = Γ ( z τ ) G(z | τ). (10) The expressions for a(τ ) and b(τ ) in the case of �(τ ) > 0 are stated in Lemma A.3 in the appendix. This function has the remarkable property that it satisfies the second functional equation G(z+ τ | τ) = (2π) τ−12 τ−z+ 12 Γ (z)G(z | τ). (11) G(z | τ) is an entire function of z with no poles and roots at z= −(mτ + n), m,n= 0,1,2, . . . . Theorem 2.4. The functionM(q | μ, λ1, λ2) satisfies M(q | μ, λ1, λ2) = Γ −q(1− 1/τ) G(1+ τ(1+ λ1) | τ)G(1− q + τ(1+ λ1) | τ) G(1+ τ(1+ λ2) | τ) G(1− q + τ(1+ λ2) | τ) × G(1+ τ | τ) G(−q + τ | τ) G(2− 2q + τ(2+ λ1 + λ2) | τ) G(2− q + τ(2+ λ1 + λ2) | τ) . (12) � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3995 Corollary 2.5. The functionM(q | μ, λ1, λ2) satisfies M(q | μ, λ1, λ2) = (2π)−1−(2+λ1+λ2)τ/2+q21+2S0(2−2q+(2+λ1+λ2)τ )Γ −q(1− 1/τ) × G(1+ τ(1+ λ1) | τ) G(1− q + τ(1+ λ1) | τ) G(1+ τ(1+ λ2) | τ) G(1− q + τ(1+ λ2) | τ) G(1+ τ | τ) G(−q + τ | τ) ×G(1− q + (2+ λ1 + λ2)τ/2 | τ) G(1/2 | τ) G(3/2− q + (2+ λ1 + λ2)τ/2 | τ) G(τ/2 | τ) ×G(1− q + (3+ λ1 + λ2)τ/2 | τ) G((1+ τ)/2 | τ) G(3/2− q + (3+ λ1 + λ2)τ/2 | τ) G(2− q + (2+ λ1 + λ2)τ | τ) , (13) where the function 2S0(z) is defined by Barnes [11] to be 2S0(z)� z2 − z(1+ τ) 2τ . (14) � Theorem 2.6. The functionM(q | μ, λ1, λ2) satisfies the functional equations M(q | μ, λ1, λ2) =M(q− τ | μ, λ1, λ2)τ (2π)τ−1Γ −τ ( 1− 1 τ ) Γ (τ − q) ×Γ ((1+ λ1)τ − (q−1))Γ ((1+ λ2)τ − (q−1)) Γ ((2+ λ1 + λ2)τ − (2q−2)) Γ ((2+ λ1 + λ2)τ − (q−2)) Γ ((3+ λ1 + λ2)τ − (2q−2)) , (15) M(q | μ, λ1, λ2) =M(q − 1 | μ, λ1, λ2)Γ (1− q/τ)Γ (2+ λ1 + λ2 − (q − 2)/τ) Γ (1− 1/τ) × Γ (1+ λ1 − (q − 1)/τ)Γ (1+ λ2 − (q − 1)/τ) Γ (2+ λ1 + λ2 − (2q − 2)/τ)Γ (2+ λ1 + λ2 − (2q − 3)/τ) . (16) � Theorem 2.7. The function logM(q | μ, λ1, λ2) has the asymptotic expansion as μ → +0 logM(q | μ, λ1, λ2) ∼ q log ( Γ (1+ λ1)Γ (1+ λ2) Γ (2+ λ1 + λ2) ) + ∞∑ r=0 (μ 2 )r+1 1 r + 1 × [ −ζ(r + 1)q + (ζ(r + 1,1+ λ1) + ζ(r + 1,1+ λ2)) ( Br+2(q) − Br+2 r + 2 ) + ζ(r + 1) × ( Br+2(q + 1) − Br+2 r + 2 ) − ζ(r + 1,2+ λ1 + λ2) ( Br+2(2q − 1) − Br+2(q − 1) r + 2 )] . (17) � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 3996 D. Ostrovsky Theorem 2.8. The function q→M(iq | μ, λ1, λ2), q ∈ R, τ > 1, is the Fourier transform of an infinitely divisible probability distribution on R with the Le´vy–Khinchine decomposi- tion logM(iq | μ, λ1, λ2) = iqm(μ) − 12q2σ 2(μ) + ∫ R\{0}(e iqu − 1− iqu/(1+ u2))dM(μ,λ1,λ2)(u) for some m(μ) ∈ R and the following gaussian component and spectral function σ 2(μ) = 4 log 2 τ , (18) M(μ,λ1,λ2)(u) = − ∫∞ u [ (ex + e−xτλ1 + e−xτλ2 + e−x(1+τ(1+λ1+λ2))) (ex − 1)(exτ − 1) − e−x(1+τ(1+λ1+λ2))/2 (ex/2 − 1)(exτ/2 − 1) ] dx x (19) for u> 0, andM(μ,λ1,λ2)(u) = 0 for u< 0. � Corollary 2.9. The probability distribution in Theorem 2.8 has a bounded, continuous, zero-free density function f(μ,λ1,λ2)(x), x∈ R. The function q→M(q | μ, λ1, λ2) for �(q) < τ and τ > 1 is the Mellin transform of the probability density function f(μ,λ1,λ2)(log y)/y, y∈ R+. � Denote the positive random variable corresponding to f(μ,λ1,λ2)(log y)/y by M(μ,λ1,λ2). Corollary 2.10. The positive and negative integral moments of M(μ,λ1,λ2) are given in Equations (6) and (7), respectively, and the asymptotic expansion of its Mellin transform is given in Equation (17). The Stieltjes moment problems for M(μ,λ1,λ2) and M −1 (μ,λ1,λ2) are indeterminate. (Determinate means that the solution to the moment problem is uniquely determined by the moments.) The left tail of M(μ,λ1,λ2) is lognormal, the right tail is a power law and its asymptotic behavior is P[M(μ,λ1,λ2) >u]∼ constu −2/μ as u→ +∞. � In summary, Theorem 2.2 shows that M(q | μ, λ1, λ2) reproduces Selberg’s for- mula for positive integral q and gives a novel formula that has a structure, which is sim- ilar to Selberg’s product, for negative integral q. Theorem 2.4 shows thatM(q | μ, λ1, λ2) is a meromorphic function of q ∈ C. Moreover, it is defined for all complex τ except τ ≤ 1 or, equivalently, except μ ≤ 0 and μ ≥ 2. Theorem 2.6 gives the functional equations of M(q | μ, λ1, λ2). It is worth pointing out that Equation (16) follows directly for integral q from Selberg’s formula. This observation was the starting point of Fyodorov et al. [20], who constructed a function that satisfies Equation (16) for complex q. Theorem 2.7 gives the desired asymptotic expansion in Equation (3) as an asymptotic series can be at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3997 exponentiated (cf. [27, Section 65]). In fact, we derived Equation (12) by summing the divergent series in Equation (17) using Hardy’s moment constant method and Ramanu- jan’s generalization of Watson’s lemma. Theorem 2.8 gives Equation (2). Remarkably, the corresponding probability distribution is of the form of the sum of an indepen- dent gaussian and a (nongaussian) infinitely divisible random variable. Corollary 2.9 says that M(q | μ, λ1, λ2) is the Mellin transform of the exponential of the distribu- tion in Theorem 2.8, which we called M(μ,λ1,λ2) so that M(μ,λ1,λ2) in law= LμN(μ,λ1,λ2), where log Lμ and log N(μ,λ1,λ2) are the gaussian and infinitely divisible components, respec- tively. Corollary 2.10 describes some of the basic properties of M(μ,λ1,λ2) and shows that it satisfies Equations (1)–(3). We conjecture that M(μ,λ1,λ2) equals ∫1 0 s λ1(1− s)λ2 dMμ(s). We will end this section with two additional remarks. First, it is clear from Theorem 2.4 that the structure of M(q | μ, λ1, λ2) has to do with ratios of the G-factors. Using Lemma 5.1, it is easy to compute the Taylor coefficients of the logarithm of such ratios ∂n ∂qn |q=0 log G(1+ a+ τ | τ)G(1− q + a+ τ | τ) = (n− 1)! ∞∑ k=1 ζ(n,1+ a+ kτ), n≥ 3. (20) Hence, the “core” of the structure of M(q | μ, λ1, λ2) is in infinite sums over values of the Hurwitz zeta function. (The sum in Equation (20) is easily seen to be a partic- ular value of the Barnes double zeta function.) Second, Corollary 2.5 is most useful for determining the roots and poles of M(q | μ, λ1, λ2). Its probabilistic interpretation is given in Theorem 3.7. Interestingly, the multiplicities of roots and orders of poles of M(q | μ, λ1, λ2) depend on certain arithmetic properties of μ, λ1, and λ2. This phe- nomenon was observed and studied in a different context in [22, 28]. 3 Probabilistic Structure ofM(q | μ, λ1, λ2) In this section, we will introduce two probabilistic decompositions of M(μ,λ1,λ2) that describe its fine structure. They are motivated by Theorem 2.8. The proofs are deferred to Section 5. We begin by noticing that Theorem 2.8 can be formulated as the equality in law of random variables M(μ,λ1,λ2) in law= Lμ N(μ,λ1,λ2), (21) where log Lμ is gaussian with zero mean and variance given in Equation (18), log N(μ,λ1,λ2) is infinitely divisible with the spectral function given in Equation (19), and Lμ and at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 3998 D. Ostrovsky N(μ,λ1,λ2) are independent. This is the first decomposition. We now proceed to describe some of the properties of N(μ,λ1,λ2). Theorem 3.1. Let τ > 1, then the asymptotic behavior ofM(q | μ, λ1, λ2) is M(−q | μ, λ1, λ2) = exp ( 2q2 log 2 τ + q logq τ + O(q) ) , q→ ∞, |arg(q)| < π. (22) � Corollary 3.2. The series g(μ,λ1,λ2)(x)� ∞∑ n=0 (−x)n n! e−2n 2 log 2/τ n−1∏ k=0 Γ (2+ λ1 + λ2 + (n+ 2+ k)/τ)Γ (1− 1/τ) Γ (1+ λ1 + (k+ 1)/τ)Γ (1+ λ2 + (k+ 1)/τ)Γ (1+ k/τ) (23) is absolutely convergent for all x∈ C. � Corollary 3.3. Given 0< �(q) < τ , ∫∞ 0 xq−1g(μ,λ1,λ2)(x)dx= Γ (q) e−2q 2 log 2/τM(q | μ, λ1, λ2). (24) � Corollary 3.4. The Laplace transform of N−1(μ,λ1,λ2) is E[e−x N −1 (μ,λ1 ,λ2) ]= g(μ,λ1,λ2)(x), x> 0. (25) In particular, g(μ,λ1,λ2)(x) > 0 for all x> 0. The Stieltjes moment problem for N −1 (μ,λ1,λ2) is determinate. � We now proceed to describe the fine structure of N(μ,λ1,λ2). Before we can formu- late our result, we need to introduce a new class of probability distributions, which is central to the structure of N(μ,λ1,λ2). Theorem 3.5. Given b, c,d, and τ such that b> −1, b+ c> −1, b+ d> −1, b+ c+ d> −1, cd> 0, and τ > 0, then, for �(q) < τ, we have the identity exp (∫∞ 0 dx x e−bx(1− e−cx)(1− e−dx) (ex − 1)(exτ − 1) (e xq − 1) ) = G(1+ b+ τ | τ) G(1− q + b+ τ | τ) G(1− q + b+ c+ τ | τ) G(1+ b+ c+ τ | τ) ×G(1− q + b+ d+ τ | τ) G(1+ b+ d+ τ | τ) G(1+ b+ c+ d+ τ | τ) G(1− q + b+ c+ d+ τ | τ) . (26) � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 3999 Denote the meromorphic function of q ∈ C given by the right-hand side of Equation (26) by N(q | τ,b, c,d). Corollary 3.6. The function q→N(q | τ,b, c,d) is the Mellin transform of a positive probability distribution X(τ,b, c,d) such that E[Xq(τ,b, c,d)]=N(q | τ,b, c,d) for �(q) < τ and log X(τ,b, c,d) is a positive, absolutely continuous, and infinitely divisible distribu- tion having the spectral function − ∫∞u e−bx(1− e−cx)(1− e−dx)/(ex − 1)(exτ − 1)dx/x for u> 0 and zero for u< 0. � Let X1, X2, and X3 be random variables having the distribution in Corollary 3.6 with the following parameters: X1 � X(τ, τλ1, τ (λ2 − λ1)/2, τ (λ2 − λ1)/2), X2 � X(τ, τ (λ1 + λ2)/2, 12 , τ/2), X3 � X(τ,0, (1+ τ + τλ1 + τλ2)/2, (1+ τ + τλ1 + τλ2)/2). (27) Given τ > 0, let Y be the positive distribution having the probability density τy−1−τ exp(−y−τ ), y> 0. Clearly, E[Yq]= Γ ( 1− q τ ) , �(q) < τ, (28) so that logY is infinitely divisible. Then, the distribution N(μ,λ1,λ2) in Equation (21) has the following decomposition. Theorem 3.7. Let τ = 2/μ, τ > 1, and let X1, X2, X3, and Y be independent. Then, there holds the equality in distribution N(μ,λ1,λ2) in law= 2π2−[3(1+τ)+2τ(λ1+λ2)]/τΓ (1− 1/τ)−1X1X2X3Y. (29) � This is the second decomposition. Remark 3.8. It is easy to see by expressing the decompositions in Equations (21) and (29) in terms of the Mellin transform that they correspond to the representation ofM(q | μ, λ1, λ2) in Corollary 2.5. � The function N(q | τ,b, c,d) has the following properties that in many ways mir- ror those ofM(q | μ, λ1, λ2). at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4000 D. Ostrovsky Corollary 3.9. N(q | τ,b, c,d) satisfies the functional equations: N(q | τ,b, c,d) =N(q − 1 | τ,b, c,d)Γ (1+ (1+ b− q)/τ) Γ (1+ (1+ b+ c+ d− q)/τ) Γ (1+ (1+ b+ c− q)/τ)Γ (1+ (1+ b+ d− q)/τ) , (30) N(q | τ,b, c,d) =N(q − τ | τ,b, c,d)Γ (1− q + b+ τ) Γ (1− q + b+ c+ d+ τ) Γ (1− q + b+ c+ τ)Γ (1− q + b+ d+ τ) . (31) � Corollary 3.10. The positive and negative integral moments l = 1,2,3, . . . of X(τ,b, c,d) are, respectively, N(q= l | τ,b, c,d) = l−1∏ k=0 Γ (1+ (b− k)/τ)Γ (1+ (b+ c+ d− k)/τ) Γ (1+ (b+ c− k)/τ)Γ (1+ (b+ d− k)/τ) , l < τ, (32) N(q= −l | τ,b, c,d) = l−1∏ k=0 Γ (1+ (1+ b+ c+ k)/τ)Γ (1+ (1+ b+ d+ k)/τ) Γ (1+ (1+ b+ k)/τ)Γ (1+ (1+ b+ c+ d+ k)/τ) . (33) � Theorem 3.11. The asymptotic behavior of N(q | τ,b, c,d) is N(−q | τ,b, c,d) = exp ( −cd τ logq + O(1) ) , q→ ∞, |arg(q)| < π. (34) � Corollary 3.12. The series h(τ,b,c,d)(x)� ∞∑ n=0 (−x)n n! n−1∏ k=0 Γ (1+ (1+ b+ c+ k)/τ)Γ (1+ (1+ b+ d+ k)/τ) Γ (1+ (1+ b+ k)/τ)Γ (1+ (1+ b+ c+ d+ k)/τ) (35) is absolutely convergent for all x∈ C. � Corollary 3.13. Given 0< �(q) < τ , ∫∞ 0 xq−1h(τ,b,c,d)(x)dx= Γ (q)N(q | τ,b, c,d). (36) � Corollary 3.14. The Laplace transform of X−1(τ,b, c,d) is E[e−xX −1(τ,b,c,d)]= h(τ,b,c,d)(x), x> 0. (37) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4001 In particular, h(τ,b,c,d)(x) > 0 for all x> 0. The Stieltjes moment problem for X−1(τ,b, c,d) is determinate. � In summary, Theorems 3.1 and 3.11 allow us to introduce two novel power series formed from the negative moments of N(μ,λ1,λ2) and X(τ,b, c,d), respectively, relate them to M(q | μ, λ1, λ2) and N(q | τ,b, c,d) by Ramanujan’s Master Theorem, and prove their positivity as corollaries of Theorems 2.8 and 3.5. Quite interestingly, these series give us the Laplace transform of N−1(μ,λ1,λ2) and X −1(τ,b, c,d), respectively, and show that both N−1(μ,λ1,λ2) and X −1(τ,b, c,d) have the determinate Stieltjes moment problem. 4 Intermittency Differentiation and the Selberg Integral In this section, we will give a derivation of the expansion in Equation (3) using our tech- nique of intermittency expansions. The rule of intermittency differentiation and ensuing intermittency expansions that we established in [37] do not apply directly as the struc- ture of the random variable ∫1 0 s λ1(1− s)λ2 dMμ(s) is more complex than what was consid- ered in our previous work. Nonetheless, our original approach does naturally generalize to the problem at hand so that we are able to derive a new intermittency expansion by retracing the original steps almost verbatim. For this reason, we will restrict ourselves to simply stating the main intermediary steps and refer the reader to [37–39] for the complete detail. We begin with the proof of Equation (1). Recall the definition of the limit log- normal measure following [6, 35]. Let ωε(s) be a gaussian process in s, whose mean and covariance are functions of a finite scale ε > 0. We consider the process Mε(t) = ∫ t 0 eωε(s) ds. (38) The mean and covariance of ωε(s)�ωμ,L ,ε(s) are defined to be E[ωε(t)]= −μ2 ( 1+ log L ε ) , (39a) Cov[ωε(t), ωε(s)]= μ log L|t− s| , ε ≤ |t− s| ≤ L , (39b) Cov[ωε(t), ωε(s)]= μ ( 1+ log L ε − |t− s| ε ) , |t− s| < ε, (39c) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4002 D. Ostrovsky and covariance is zero in the remaining case of |t− s| ≥ L. Thus, ε is used as a truncation scale. L is the fundamental decorrelation length of the process that is from now on set to L = 1. μ is the intermittency parameter. Note that E[exp(ωε(s))]= 1 so that E[Mε(t)]= t. The interest in the limit lognormal construction stems from the ε → 0 limit. Using the theory of T-martingales developed by Kahane in a series of papers [23–25], and the work of Barral and Mandelbrot [12] on log-Poisson cascades, Bacry and Muzy [8] showed that Mε(t) converges weakly (as a measure on R+) a.s. to a limit process Mμ(t) = limε→0Mε(t) provided 0≤ μ < 2, and the limit is nondegenerate in the sense that E[Mμ(t)]= t. Proof of Equation (1). Consider the positive integral moments of ∫1 0 ϕ(s)dMμ(s) for some positive function ϕ(s) such that ∫1 0 ϕ(s)ds exists. Let ε = 1/N, ω j �ωε(sj), sj = εj, j = 1, . . . , N. As ω1, . . . , ωN are jointly gaussian, we have E[ε N∑ j=1 ϕ(sj) e ω j ]l = εl N∑ j1... jl=1 ϕ(sj1) · · ·ϕ(sjl ) emean(ω j1+···+ω jl )+ 1 2 var(ω j1+···+ω jl ). (40) It follows from Equations (39a) and (39b) that E[ε N∑ j=1 ϕ(sj) e ω j ]l = εl N∑ j1... jl=1 ϕ(sj1) · · ·ϕ(sjl ) l∏ k Selberg Integral as a Meromorphic Function 4003 v(μ, f, F ) = limε→0 vε(μ, f, F ) and vε(μ, f, F )�E[F ( ∫1 0 e μf(s)ϕ(s)dMε(s))] with dMε(s) as in Equation (38). Also, let g(s1, s2) be defined by g(s1, s2)�− log |s1 − s2|. (44) Finally, we will use [0,1]k to denote the k-dimensional unit cube [0,1]× · · · × [0,1]. Then, we have the following rule of intermittency differentiation. Theorem 4.1. Let μ < 1. The expectation v(μ, f, F ) is invariant under intermittency dif- ferentiation and satisfies ∂ ∂μ v(μ, f, F ) = ∫ [0,1] v(μ, f + g(·, s), F (1)) eμf(s) f(s)ϕ(s)ds+ 1 2 ∫ [0,1]2 v(μ, f + g(·, s1) + g(·, s2), F (2)) eμ( f(s1)+ f(s2)+g(s1,s2))g(s1, s2)ϕ(s1)ϕ(s2)d s(2). (45) � The mathematical content of Equation (45) is that differentiation with respect to the intermittency parameter μ is equivalent to a combination of two functional shifts induced by the g function. It is clear that both terms in Equation (45) are of the same functional form as the original functional in Equation (43) so that Theorem 4.1 allows us to compute derivatives of all orders. There results the following formal expansion with some coefficients Hn,k(ϕ) that are independent of F . Let Sl(μ)� ∫ [0,1]l l∏ i=1 ϕ(si) l∏ i< j |si − sj|−μ ds1 · · ·dsl, (46) x� ∫1 0 ϕ(s)ds. (47) Theorem 4.2. The expectation E[F ( ∫1 0 ϕ(s)dMμ(s))] has the intermittency expansion E [ F (∫1 0 ϕ(s)dMμ(s) )] = F (x) + ∞∑ n=1 μn n! [ 2n∑ k=2 F (k)(x)Hn,k(ϕ) ] . (48) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4004 D. Ostrovsky The expansion coefficients Hn,k(ϕ) are given by the binomial transform of the derivatives of the positive integral moments Hn,k(ϕ) = (−1) k k! k∑ l=2 (−1)l ( k l ) xk−l ∂nSl ∂μn |μ=0, (49) and satisfy the identity Hn,k(ϕ) = 0 ∀k> 2n. (50) � Remark 4.3. Equation (42) and Theorem 4.2 say that the intermittency expansion in Equation (48) is an exactly renormalized expansion in the centered moments of ∫1 0 ϕ(s)dMμ(s). Indeed, we have the identity ∂ n/∂μn |μ=0 E[( ∫1 0 ϕ(s)dMμ(s) − x)k]= k! Hn,k(ϕ). � From now on, we will focus on ϕ(s)� sλ1(1− s)λ2 , λ1, λ2 > −μ2 , (51) which corresponds to the full Selberg integral, and write Hn,k(λ1, λ2) for the expansion coefficients. Clearly, we have x= Γ (1+ λ1)Γ (1+ λ2) Γ (2+ λ1 + λ2) . (52) The moments Sl(μ) = Sl(μ, λ1, λ2) are given by Selberg’s product in Equation (1). By expanding log Sl(μ, λ1, λ2) in powers of μ near zero, we obtain log Sl(μ, λ1, λ2) = l log(x) + ∞∑ p=1 μpcp(l, λ1, λ2), (53) cp(l, λ1, λ2) = 12pp ⎡ ⎣(ζ(p,1+ λ1) + ζ(p,1+ λ2)) l−1∑ j=0 j p + ζ(p) l−1∑ j=0 ( j + 1)p − ζ(p)l − ζ(p,2+ λ1 + λ2) l−1∑ j=0 (l + j − 1)p ⎤ ⎦ . (54) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4005 Let ζ(1,a)�−ψ(a). The expansion coefficients satisfy the following key recurrence rela- tion, which determines Hn,k(λ1, λ2) uniquely as H1,k(λ1, λ2) = xkA0,k. Proposition 4.4. Hn+1,k(λ1, λ2) = xkAn,k + n−1∑ r=0 ( n r ) k∑ t=2 xk−tHn−r,t(λ1, λ2)Br,t,k, n≥ 0,k≥ 2, (55) An,k� (−1)k (n+ 1)!k! k∑ l=2 (−1)l ( k l ) cn+1(l, λ1, λ2), (56) Br,t,k� (−1)kt! (r + 1)!k! k∑ l=t (−1)l ( k l )( l t ) cr+1(l, λ1, λ2). (57) � Wewill use Proposition 4.4 to compute the intermittency expansion of the Mellin transform. The Mellin transform corresponds to F (x) = xq in Equation (48) for some fixed q ∈ C, so that we obtain the formal expansion E [(∫1 0 sλ1(1− s)λ2 dMμ(s) )q] = xq + ∞∑ n=1 μn n! fn(q, λ1, λ2), (58) fn(q, λ1, λ2) = ∞∑ k=2 (q)kx q−kHn,k(λ1, λ2), n= 1,2,3, . . . , (59) where the sum in Equation (59) has been extended to infinity by Theorem 4.2 and (q)k� q(q − 1)(q − 2) · · · (q − k+ 1). Define the coefficients br(q, λ1, λ2), r = 0,1,2 . . . br(q, λ1, λ2)� 1 2r+1 [ (ζ(r + 1,1+ λ1) + ζ(r + 1,1+ λ2)) ( Br+2(q) − Br+2 r + 2 ) − ζ(r + 1)q + ζ(r + 1) ( Br+2(q + 1) − Br+2 r + 2 ) − ζ(r + 1,2+ λ1 + λ2) ( Br+2(2q − 1) − Br+2(q − 1) r + 2 )] . (60) Given Equations (54), (56), and (57), a direct computation results in the identities ∞∑ k=2 (q)kAn,k =n!bn(q, λ1, λ2), ∞∑ k=t (q)kBr,t,k = (q)tr!br(q, λ1, λ2). (61) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4006 D. Ostrovsky Finally, using Equation (55) and the recurrence relation of complete, exponential Bell polynomials, we arrive at the formal solution for the intermittency expansion of the Mellin transform. Theorem 4.5. Let f0(q, λ1, λ2) = xq. The expansion coefficients of the Mellin transform satisfy fn+1(q, λ1, λ2) =n! n∑ r=0 fn−r(q, λ1, λ2) (n− r)! br(q, λ1, λ2), (62) E [(∫1 0 sλ1(1− s)λ2 dMμ(s) )q] = xq exp ( ∞∑ r=0 μr+1 r + 1br(q, λ1, λ2) ) . (63) � This completes our derivation of Equation (3). The infinite series in Equation (63) is generally divergent and so must be treated as defining the asymptotic expansion of the Mellin transform in the limit of small intermittency. Consider the normalized random variable M˜μ(λ1, λ2)� Γ (2+ λ1 + λ2) Γ (1+ λ1)Γ (1+ λ2) ∫1 0 sλ1(1− s)λ2 dMμ(s). (64) Corollary 4.6. Given constants a and s and a smooth function F (s), the intermittency expansion of the general transform of log M˜μ(λ1, λ2) E[F (s+ a log M˜μ(λ1, λ2))]= ∞∑ n=0 Fn(a, s, λ1, λ2) μn n! (65) is determined by F0(a, s, λ1, λ2) = F (s), Fn+1(a, s, λ1, λ2) = n∑ r=0 n! (n− r)!br ( a d ds , λ1, λ2 ) Fn−r(a, s, λ1, λ2). (66) � Corollary 4.6 shows that the solution for the general transform in Equation (65) is obtained by replacing q with ad/ds in the solution for the Mellin transform in Theorem 4.5. The general transform is particularly interesting in the special case of purely imaginary a, in which case the operator F (s) →E[F (s+ a log M˜μ(λ1, λ2))] is self- adjoint. at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4007 5 Proofs In this section, we will give proofs of all results that were stated in Sections 2 and 3. All the proofs with the exception of the proof of Theorem 2.2 rely on various properties of the Alexeiewsky–Barnes G-function, which will be reviewed in the appendix. Recall that τ = 2/μ as defined in Section 2. Proof of Theorem 2.2. Recall the identity (cf. [49, Section 12.13]), ∞∏ m=1 ⎡ ⎣ k∏ j=1 m− aj m− bj ⎤ ⎦= k∏ j=1 Γ (1− bj) Γ (1− aj) (67) that holds for arbitrary constants aj and bj so long as ∑ j aj = ∑ j bj. Given q= l, l = 1,2,3 . . . , using the functional equation of the gamma function, we obtain for the infinite product in Equation (5) ∞∏ m=1 (mτ)2l Γ (1− l +mτ) Γ (1+mτ) Γ (1− l + τλ1 +mτ) Γ (1+ τλ1 +mτ) Γ (1− l + τλ2 +mτ) Γ (1+ τλ2 +mτ) × Γ (2− l + τ(λ1 + λ2) +mτ) Γ (2− 2l + τ(λ1 + λ2) +mτ) = ∞∏ m=1 m2l l−1∏ j=0 (λ1 + λ2 +m− (l + j − 1)/τ) (m− (l − j − 1)/τ) × 1 (λ1 +m− (l − j − 1)/τ)(λ2 +m− (l − j − 1)/τ) . (68) It is easy to see that the condition ∑ j aj = ∑ j bj is fulfilled with k= 3l factors so that ∞∏ m=1 (mτ)2l Γ (1− l +mτ) Γ (1+mτ) Γ (1− l + τλ1 +mτ) Γ (1+ τλ1 +mτ) Γ (1− l + τλ2 +mτ) Γ (1+ τλ2 +mτ) × Γ (2− l + τ(λ1 + λ2) +mτ) Γ (2− 2l + τ(λ1 + λ2) +mτ) = l−1∏ j=0 Γ (1− j/τ)Γ (1+ λ1 − j/τ)Γ (1+ λ2 − j/τ) Γ (1+ λ1 + λ2 − (l + j − 1)/τ) . (69) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4008 D. Ostrovsky On the other hand, using the functional equation of the gamma function, we have τ l Γ (2− 2l + τ(1+ λ1 + λ2)) Γ (2− l + τ(1+ λ1 + λ2)) = l−1∏ j=0 1 1+ λ1 + λ2 − (l + j − 1)/τ . (70) Thus, M(l | μ, λ1, λ2) = Γ (1− l/τ) Γ l(1− 1/τ) l−1∏ j=0 Γ (1− j/τ)Γ (1+ λ1 − j/τ)Γ (1+ λ2 − j/τ) Γ (2+ λ1 + λ2 − (l + j − 1)/τ) . (71) It is elementary now to see that this expression is the same as Equation (6). The proof of Equation (7) goes through verbatim. Instead of Equations (69) and (70), we obtain ∞∏ m=1 (mτ)−2l Γ (1+ l +mτ) Γ (1+mτ) Γ (1+ l + τλ1 +mτ) Γ (1+ τλ1 +mτ) Γ (1+ l + τλ2 +mτ) Γ (1+ τλ2 +mτ) × Γ (2+ l + τ(λ1 + λ2) +mτ) Γ (2+ 2l + τ(λ1 + λ2) +mτ) = l−1∏ j=0 Γ (1+ λ1 + λ2 + (l + j + 2)/τ) Γ (1+ ( j + 1)/τ) 1 Γ (1+ λ1 + ( j + 1)/τ)Γ (1+ λ2 + ( j + 1)/τ) . (72) τ−l Γ (2+ 2l + τ(1+ λ1 + λ2)) Γ (2+ l + τ(1+ λ1 + λ2)) = l−1∏ j=0 (1+ λ1 + λ2 + (l + j + 2)/τ). (73) Thus, we finally obtain M(−l | μ, λ1, λ2) = Γ (1+ l/τ) Γ −l(1− 1/τ) l−1∏ j=0 Γ (2+ λ1 + λ2 + (l + j + 2)/τ) Γ (1+ ( j + 1)/τ) × 1 Γ (1+ λ1 + ( j + 1)/τ)Γ (1+ λ2 + ( j + 1)/τ) , (74) which is the same as Equation (7). � Proof of Theorem 2.4. The starting point is Shintani’s identity [44], which is stated here in a form that is equivalent to the original formulation given in the appendix in at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4009 Equation (A.30). Given z∈ C and |arg(τ )| < π, G(z+ τ | τ) = (2π) (τ−1)2 τ− 12 eγ (z−z 2) 2τ ∞∏ m=1 (mτ)z−1e (z2−z) 2mτ Γ (1+mτ) Γ (z+mτ) . (75) Using the functional equations of the Alexeiewsky–Barnes G-function, we first reduce some of the G-factors in Equation (12) as follows: 1 G(−q + τ | τ) G(2− 2q + τ(2+ λ1 + λ2) | τ) G(2− q + τ(2+ λ1 + λ2) | τ) = τqΓ (2− 2q + τ(1+ λ1 + λ2)) Γ (2− q + τ(1+ λ1 + λ2)) Γ (1− q/τ)) G(1− q + τ | τ) G(2− 2q + τ(1+ λ1 + λ2) | τ) G(2− q + τ(1+ λ1 + λ2) | τ) . (76) We now apply Shintani’s identity to each of the resulting G-factors in Equation (12). It is easy to see that the terms that are quadratic in q all cancel out resulting in Equation (5). � Proof of Corollary 2.5. This result follows from Theorem 2.4 by means of the following doubling formula: G(2z | τ) = (2π)−z21+2S0(2z)G(z | τ)G(z+ 1/2 | τ)G(z+ τ/2 | τ)G(z+ (1+ τ)/2 | τ) G(1/2 | τ)G(τ/2 | τ)G((1+ τ)/2 | τ) . (77) This formula is a special case of the general multiplication formula that Barnes [11] derived for the double gamma function (cf. Equation (A.4) in the appendix). � Proof of Theorem 2.6. The functional equations in Equations (15) and (16) are immedi- ate corollaries of the functional equations of G(z | τ). � Proof of Theorem 2.7. The key element in the proof is the following integral, which generalizes the integral that we introduced in [38]. Let �(q) < 1+ a+ �(τ ), a> −�(τ ), and �(τ ) > 0. I (q | a, τ )� ∫∞ 0 dx x e−ax exτ − 1 [ exq − 1 ex − 1 − q − (q2 − q) 2 x ] . (78) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4010 D. Ostrovsky This integral is the essential link that connects the asymptotic series in Equation (3) and the Alexeiewsky–Barnes G-function. In fact, on the one hand, I (q | a, τ ) has the asymp- totic expansion in the limit τ → +∞ I (q | aτ, τ ) ∼ ∞∑ r=1 ζ(r + 1,1+ a) r + 1 ( Br+2(q) − Br+2 r + 2 )/ τ r+1. (79) This follows from a slight extension of Ramanujan’s generalization of Watson’s lemma (cf. Lemma 10.2 in Chapter 38 of [14]). Given a function f(y) that has the asymptotic expansion f(y) ∼ ∑∞ r=1 ary r as y→ 0 and a> −1, then as τ → +∞, ∫∞ 0 e−ayτ eyτ − 1 f(y)dy∼ ∞∑ r=1 ζ(r + 1,1+ a)r!ar/τ r+1. (80) Equation (79) follows from Equation (80) using f(y)� 1 y [ eyq − 1 ey − 1 − q − (q2 − q) 2 y ] , (81) whose asymptotic expansion we derived in [38]. The proof of Equation (80) is quite sim- ilar to the proof of Ramanujan’s lemma in [14] and will be omitted. On the other hand, I (q | a, τ ) is related to the G-function by the following: Lemma 5.1. Given �(q) < 1+ a+ �(τ ), a> −�(τ ), and �(τ ) > 0, I (q | a, τ ) = log G(1+ a+ τ | τ) G(1− q + a+ τ | τ) − q log [ Γ ( 1+ a τ )] + (q 2 − q) 2τ ψ ( 1+ a τ ) . (82) � The proof is given in the appendix. The only other term in Equation (17), which has a structure that is different from the series in Equation (79), can be treated using the elementary identity ∞∑ r=1 ζ(r + 1) r + 1 / τ r+1 = logΓ (1− 1/τ) + ψ(1) τ , |τ | > 1. (83) The result now follows by a straightforward algebraic reduction. � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4011 Remark 5.2. The relationship between the G-function and the asymptotic series in Equation (79) can also be derived directly from the asymptotic expansion of the dou- ble gamma function given in [33]. � The proof of Theorem 2.8 is based on the Malmste´n-type formula for logG(z | τ) that was first derived by Lawrie and King [29] using the functional equations of G(z | τ). We will give its elementary proof in the appendix. Given �(z),�(τ ) > 0, logG(z | τ) = ∫∞ 0 dt t [ 1− z etτ − 1 + (1− z) e −tτ + (z2 − z)e −tτ 2τ + 1− e −t(z−1) (et − 1)(1− e−tτ ) ] . (84) Proof of Theorem 2.8. Let �(q) < �(τ ), 0< �(1/τ) < 1, and λ1, λ2 > −μ/2. Applying Equation (84) to each G-factor in Theorem 2.4, we can write logM(q | μ, λ1, λ2) = ∫∞ 0 dt t [ 1 (et − 1)(etτ − 1) (e t(q−λ1τ) + et(q−λ2τ) + et(q+1) +et(q−1−τ(1+λ1+λ2)) − et(2q−1−τ(1+λ1+λ2))) + A(t) + B(t)q + C (t)q2 ] (85) for some functions A(t), B(t), and C (t) depending also on τ , λ1, and λ2. Note that C (t) = 0 as it is proportional to 22 − 2− 1− 1. Then, we can rewrite this expression in the form logM(q | μ, λ1, λ2) = ∫∞ 0 dt t [ (etq − 1− tq) (et − 1)(etτ − 1) (e −tλ1τ + e−tλ2τ + et + e−t(1+τ(1+λ1+λ2))) − 1 2 (2tq)2 e−t(1+τ(1+λ1+λ2)) (et − 1)(etτ − 1) + A(t) + B(t)q ] − ∫∞ 0 dt t [ (e2tq − 1− 2tq − (2tq)2/2) (et − 1)(etτ − 1) e −t(1+τ(1+λ1+λ2)) ] (86) for some appropriately modified A(t) and B(t). It follows that we have logM(q= 0 | μ, λ1, λ2) = ∫∞ 0 dt t A(t). (87) On the other hand,M(q= 0 | μ, λ1, λ2) = 1 by construction. Hence this term vanishes. We now change variables t′ = 2t in the second integral and then bring the two integrals back at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4012 D. Ostrovsky under the same integral sign. logM(q | μ, λ1, λ2) = ∫∞ 0 dt t [ (etq − 1− tq) ( (e−tλ1τ + e−tλ2τ + et + e−t(1+τ(1+λ1+λ2))) (et − 1)(etτ − 1) − e−t/2(1+τ(1+λ1+λ2)) (et/2 − 1)(etτ/2 − 1) ) + q 2t2 2 ( e−t/2(1+τ(1+λ1+λ2)) (et/2 − 1)(etτ/2 − 1) − 4 e−t(1+τ(1+λ1+λ2)) (et − 1)(etτ − 1) ) + B(t)q ] . (88) Denote f(t)� (e −tλ1τ + e−tλ2τ + et + e−t(1+τ(1+λ1+λ2))) (et − 1)(etτ − 1) − e−t/2(1+τ(1+λ1+λ2)) (et/2 − 1)(etτ/2 − 1) , (89) g(t)� t 2 (et/2 − 1)(etτ/2 − 1) − (2t)2 (et − 1)(etτ − 1) . (90) The functions f and g have the property f= O(t−1), g= O(t) as t→ 0 and f, g are exponen- tially small as t→ +∞. Noting the individual existence and equality of the integrals ∫∞ 0 dt t [ t2 (e−t/2(1+τ(1+λ1+λ2)) − 1) (et/2 − 1)(etτ/2 − 1) ] = ∫∞ 0 dt t [ (2t)2 (e−t(1+τ(1+λ1+λ2)) − 1) (et − 1)(etτ − 1) ] , (91) we can write logM(q | μ, λ1, λ2) = q ∫∞ 0 dt t B(t) + ∫∞ 0 dt t [(etq − 1− tq)f(t)]+ q 2 2 ∫∞ 0 dt t g(t). (92) Denote σ 2(μ)� ∫∞ 0 dt t g(t). (93) Define the functionM(μ,λ1,λ2)(u)�− ∫∞ u f(t)dt/t for u> 0 andM(μ,λ1,λ2)(u)� 0 for u< 0. We have thus established the decomposition for �(q) < �(τ ) logM(q | μ, λ1, λ2) = q ∫∞ 0 dt t B(t) + 1 2 q2σ 2(μ) + ∫ R\{0} (euq − 1− uq)dM(μ,λ1,λ2)(u). (94) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4013 Finally, if τ is in addition real so that τ > 1, we have the inequalities f(t) > 0 for t> 0, (95) σ 2(μ) > 0. (96) Their validity can be established as follows. Let a= e−tλ1τ/2, b= e−tλ2τ/2, c= e−t/2, and d= e−tτ/2. Note that a,b≥ 0 and 0< c,d< 1. Then, Equation (95) is equivalent to a2 + b2 + c−2 + a2b2c2d2 − ab(1+ c)(1+ d) > 0. (97) The latter inequality follows from a2 + b2 + c−2 + a2b2c2d2 − ab(1+ c)(1+ d) = (a− b)2 + c−2 + ab(1− c)(1− d) + (abcd− 1)2 − 1≥ c−2 − 1> 0. (98) The integral for σ 2(μ) can be computed explicitly ∫∞ 0 dt t g(t) = ∫∞ 0 dt t [ t2 (et/2 − 1)(etτ/2 − 1) − 4 τ e−t ] − ∫∞ 0 dt t [ (2t)2 (et − 1)(etτ − 1) − 4 τ e−t ] , = 4 τ ∫∞ 0 dt t [e−t − e−2t]= 4 τ log 2. (99) Hence σ 2(μ) > 0 and M(μ,λ1,λ2)(u) is continuous and non-decreasing on (−∞,0) and (0,∞) and satisfies the integrability and limit conditions ∫[−1,1]\{0} u2 dM(μ,λ1,λ2)(u) < ∞, limu→±∞M(μ,λ1,λ2)(u) = 0 so thatM(μ,λ1,λ2)(u) is a valid spectral function. (Note that it is only the mean ∫∞ 0 B(t)dt/t that necessitates τ > 1. The gaussian component σ 2(μ) and spectral functionM(μ,λ1,λ2)(u) satisfy the required properties for all τ > 0.) The decom- position in Equation (94) assumes the canonical form (cf. Theorem 4.4 in Chapter 4 of [46]), in the case of purely imaginary q by a trivial change in the linear term. � Proof of Corollary 2.9. This follows from Theorem 2.8 by some general properties of analytic and infinitely divisible characteristic functions. Denote the random variable corresponding to the Le´vy–Khinchine decomposition in Theorem 2.8 by logM(μ,λ1,λ2). As it has a nonzero gaussian component, it is absolutely continuous with a bounded, con- tinuous, and zero-free density by Theorem 8.4 and Corollary 8.8 in Chapter 4 of [46]. at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4014 D. Ostrovsky Denote the probability density of logM(μ,λ1,λ2) by f(μ,λ1,λ2)(x), x∈ R. The Fourier trans- form of f(μ,λ1,λ2)(x) isM(iq | μ, λ1, λ2) by construction, which is analytic as a function of q in the strip �(q) > −τ . Then, by the fundamental theorem of analytic characteristic func- tions (cf. Theorem 7.1.1 in Chapter 7 of [30]), we have for all q in the strip of analyticity �(q) > −τ M(iq | μ, λ1, λ2) = ∫ R eiqx f(μ,λ1,λ2)(x)dx. (100) On the other hand, the random variable M(μ,λ1,λ2) has the density f(μ,λ1,λ2)(log y)/y, y∈ R+ so that the right-hand side of Equation (100) is precisely its Mellin transform for�(q) < τ M(q | μ, λ1, λ2) = ∫∞ 0 yq f(μ,λ1,λ2)(log y) dy y =E[Mq(μ,λ1,λ2)] (101) as seen upon relabeling iq→ q and changing variables y= ex. � Proof of Corollary 2.10. We simply note that the spectral functionM(μ,λ1,λ2)(u) is sup- ported on R+ and decays as exp(−2u/μ)/u up to a constant in the limit u→ +∞ (cf. Equation (89)). Hence the argument that we gave in [38] in the special case of λ1 = λ2 = 0 goes through verbatim. � Proof of Theorem 3.1. The proof is based on the asymptotic formula for logG(z | τ) that was first derived by Billingham and King [15]. logG(z | τ) = z 2 2τ log(z) − z 2 τ ( 3 4 + log τ 2 ) − 1 2 ( 1 τ + 1 ) z log z + 1 2 ( log τ τ + 1 2 + log τ + 1+ log 2π ) z + ( τ 12 + 1 4 + 1 12τ ) log z+ O(1), z→ ∞, |arg(z/τ)| < π. (102) The result now follows from Theorem 2.4 by a tedious but straightforward calculation. � Proof of Corollary 3.2. Consider the coefficients of the series in Equation (23). By Theorem 2.2, we can simply write g(μ,λ1,λ2)(x) = ∞∑ n=0 (−x)n n! e−2n 2 log 2/τM(−n| μ, λ1, λ2). (103) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4015 Now, using Theorem 3.1, we obtain the estimate M(−n− 1 | μ, λ1, λ2) = exp ( 4 log 2 τ n+ logn τ + O(1) ) M(−n| μ, λ1, λ2), n→ +∞, (104) so that the ratio of the (n+ 1) st to the nth coefficient of the series behaves as |x| exp((1/τ − 1) logn+ O(1)) in the limit n→ +∞. As τ > 1, the result follows by the ratio test. � Proof of Corollary 3.3. Consider first the case of 0< �(q) < 1. In this case, the result follows directly from Ramanujan’s Master Theorem (cf. [3]), provided we can show thatM(q | μ, λ1, λ2) satisfies Hardy’s conditions. Ramanujan’s Master Theorem says that there holds the general identity of the form ∫∞ 0 xs−1 [ ∞∑ n=0 λ(n) (−x)n n! ] dx= Γ (s)λ(−s), 0< �(s) < δ, (105) provided λ(s) is analytic over �(s) ≥ −δ and satisfies the bound ∣∣∣∣ λ(s)Γ (1+ s) ∣∣∣∣< C exp(A�(s) + B|�(s)|), �(s) ≥ −δ, (106) for some constants A, B, C , and δ such that 0< δ < 1 and B < π . Let λ(s)� e−2s2 log 2/τM(−s | μ, λ1, λ2). (107) It is analytic for �(s) > −τ . By Theorem 3.1 and Stirling’s formula, we have the estimate λ(s) Γ (1+ s) = exp ( s log s ( 1 τ − 1 ) + O(s) ) , s→ ∞, |arg(s)| < π. (108) Finally, we have the identity �(s log s) = �(s) log |s| − �(s)arg(s) so that Hardy’s condition is fulfilled for any 0< δ < 1, which completes the proof for 0< �(s) < 1. The result for 0< �(s) < τ follows by analytic continuation. In fact, the right-hand side of Equation (105) is known to be analytic for 0< �(s) < τ, the left-hand side is also analytic over this domain provided we show that g(μ,λ1,λ2)(x) ∼ x −τ as x→ +∞. This follows from the proof of Corollary 3.4. � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4016 D. Ostrovsky Proof of Corollary 3.4. Recall that Theorem 2.8 can be stated in the form M(μ,λ1,λ2) in law= LμN(μ,λ1,λ2), where log Lμ is gaussian with zero mean and variance as in Equation (18) and log N(μ,λ1,λ2) is an independent infinitely divisible random variable having the spec- tral function as in Equation (19). Denote the unknown probability density function of N(μ,λ1,λ2) by h(μ,λ1,λ2)(x). It exists by [30, Theorem 5.5.7] as the spectral function of log N(μ,λ1,λ2) satisfies ∫∞ 0 dM(μ,λ1,λ2)(u) = ∞ (cf. Equation (19)). In terms of theMellin trans- form, Theorem 2.8 takes the form ∫∞ 0 tq−1[th(μ,λ1,λ2)(t)] dt= e−2q 2 log 2/τM(q | μ, λ1, λ2), �(q) < τ. (109) On the other hand, we will show that xh(μ,λ1,λ2)(x) is related to g(μ,λ1,λ2)(x) by the Mellin convolution g(μ,λ1,λ2)(x) = ∫∞ 0 th(μ,λ1,λ2)(t) e −x/tdt t , x> 0. (110) In fact, recalling the definition of Γ (q) as the Mellin transform of the exponential func- tion exp(−t), we obtain Equation (110) from Equations (24) and (109) by the equality of the Mellin transforms of the left- and right-hand sides of Equation (110) on 0< �(q) < 1. Hence g(μ,λ1,λ2)(x) is positive. Now, the probability density of N −1 (μ,λ1,λ2) is h(μ,λ1,λ2)(1/x)/x 2 so that Equation (110) is equivalent to the expression for the Laplace transform of N−1(μ,λ1,λ2) in Equation (25) as seen by a simple change of variables. By Theorem 3.1 and Equation (109), the positive integral moments of N−1(μ,λ1,λ2) satisfy E[N−n(μ,λ1,λ2)]= e−2n 2 log 2/τM(−n| μ, λ1, λ2) = exp((nlogn/τ + O(n)), n→ +∞. (111) As τ > 1, they grow slower than the moments of the exponential distribution so that the corresponding moment problem is determinate by the Carleman criterion. Finally, it is not difficult to see that h(μ,λ1,λ2)(t) ∼ t −τ−1, t→ +∞ (112) so that g(μ,λ1,λ2)(x) ∼ x −τ as x→ +∞ by Equation (110). � Proof of Theorem 3.5. It is sufficient to consider τ > 1 as both the left- and right-hand sides of Equation (26) exist for all τ > 0. By the definition of I (q | a, τ ) in Equation (78), at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4017 we can write for the integral in Equation (26) ∫∞ 0 dx x e−bx(1− e−cx)(1− e−dx) (ex − 1)(exτ − 1) (e xq − 1) = I (q | b, τ ) − I (q | b+ c, τ ) − I (q | b+ d, τ ) + I (q | b+ c+ d, τ ) + ∫∞ 0 dx x 1 (exτ − 1) [e −xb − e−x(b+c) − e−x(b+d) + e−x(b+c+d)](q + (q2 − q)x/2). (113) The integral on the right-hand side of Equation (113) can be computed using Equa- tions (A.28) and (A.29) in the appendix. The result follows from Lemma 5.1. � Proof of Corollary 3.6. The argument is similar to the proof of Corollary 2.9. The left- hand side of Equation (26) is of the Le´vy–Khinchine functional form, up to a triv- ial change in the linear term. Hence, there is a unique infinitely divisible distribu- tion log X(τ,b, c,d) such that E[exp(iq log X(τ,b, c,d))]=N(iq | τ,b, c,d) for q ∈ R. It is absolutely continuous by Theorem 5.5.7 in [30]. It is positive, that is, supported on (0,+∞), by Proposition 8.2 in Chapter 4 of [46]. Finally, by the fundamental theorem of analytic characteristic functions (cf. Theorem 7.1.1 in Chapter 7 of [30]), we have E[Xq(τ,b, c,d)]=N(q | τ,b, c,d) for �(q) < τ . � Proof of Theorem 3.7. The starting point is Equation (21) and Equations (92), (98), and (99) in the proof of Theorem 2.8. By combining these equations, we can write for �(q) < τ logE[Nq(μ,λ1,λ2)]= Cq + ∫∞ 0 dx x 1 (ex − 1)(exτ − 1) [e −xλ1τ/2 − e−xλ2τ/2]2(exq − 1) + ∫∞ 0 dx x 1 (ex − 1)(exτ − 1)e −xλ1τ/2e−xλ2τ/2(1− e−x/2)(1− e−xτ/2)(exq − 1) + ∫∞ 0 dx x 1 (ex − 1)(exτ − 1) [e −xλ1τ/2e−xλ2τ/2e−x/2e−xτ/2 − 1]2(exq − 1) + ∫∞ 0 dx x 1 exτ − 1 (e xq − 1− xq) (114) for some unknown constant C . Denote the integrals by I j, j = 1, . . . ,4. It is easy to see that the first three are of the same functional form as the integral in Theorem 3.5. Recall- ing the definition of X1, X2, and X3 in Equation (27), logE[Xqj ]= I j, j = 1,2,3. (115) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4018 D. Ostrovsky The fourth integral is given in the appendix (cf. Equation (A.28)). Recalling the definition of Y, we can write logE[Yq]= logΓ (1− q/τ) = I4 + Cq (116) for some constant C . The result now follows from Corollary 2.5 and the doubling formula (cf. Equation (77)). � Proof of Corollary 3.9. The functional equations of N(q | τ,b, c,d) follow from those of G(z | τ). � Proof of Corollary 3.10. The values of N(q | τ,b, c,d) at the integers follow from Equation (30) in Corollary 3.9. � Proof of Theorem 3.11 and Corollaries 3.12–3.14. The argument is the same as in the proof of Theorem 3.1 and Corollaries 3.2–3.4. � 6 Conclusions and Open Questions We have extended the rule of intermittency differentiation and resulting intermittency expansions to general functionals of the limit lognormal random measure of the form E[F ( ∫1 0 ϕ(s)dMμ(s))]. We showed that the intermittency expansion of such a functional is an exactly renormalized expansion in the centered moments of ∫1 0 ϕ(s)dMμ(s). In the special case of ϕ(s) = sλ1(1− s)λ2 the moments are given by the Selberg integral. We have used Selberg’s formula to compute the intermittency expansion of the Mellin transform of ∫1 0 s λ1(1− s)λ2 dMμ(s) for arbitrary λ1, λ2 > −μ/2 thereby extending our previous work in the special case of λ1, λ2 = 0. We interpret this expansion as the asymptotic expansion of the Mellin transform in the limit of small intermittency μ → 0 and show that it gives rise to the intermittency expansion of the general transform of log ∫1 0 s λ1(1− s)λ2 dMμ(s). We have introduced, described the structure, and given several representations of a meromorphic function M(q | μ, λ1, λ2) that extends the Selberg integral as a func- tion of its dimension to all complex q and complex μ except μ ≤ 0 and μ ≥ 2. This function is expressed in terms of the Alexeiewsky–Barnes G-function. Its main prop- erties are that it satisfies a pair of functional equations, has the same asymptotic expansion in the limit μ → 0 as the intermittency expansion of the Mellin transform of ∫1 0 s λ1(1− s)λ2 dMμ(s), and that for �(q) < 2/μ and 0< μ < 2 the function q→M(q | μ, λ1, λ2) is the Mellin transform of an absolutely continuous probability distribution at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4019 M(μ,λ1,λ2) on the positive real line. The logarithm of this distribution is infinitely divis- ible so that M(μ,λ1,λ2) in law= Lμ N(μ,λ1,λ2), where log Lμ is a zero-mean gaussian with vari- ance 2μ log 2 and log N(μ,λ1,λ2) is an independent (nongaussian) infinitely divisible dis- tribution. Moreover, N(μ,λ1,λ2) in law= const X1X2X3Y, where Xi belongs to a new class of distributions and is characterized by its Mellin transform, Y has a known density, and they are all independent. We have expressed the moments of N−1(μ,λ1,λ2) and X −1 i in the form of novel product formulas of Selberg type, proved that the corresponding power series are positive, and shown that the Stieltjes moment problems for N−1(μ,λ1,λ2) and X−1i , unlike those for M(μ,λ1,λ2) and M −1 (μ,λ1,λ2) , are determinate. We conjecture that M(μ,λ1,λ2) = ∫1 0 s λ1(1− s)λ2 dMμ(s). We have also reviewed the Alexeiewsky–Barnes G-function and shown that a number of its key properties follow from Barnes’s original characterization of the double gamma modular functions. Our work leads to a number of open questions that are beyond the scope of this paper. We will mention four such problems, which we consider to be the most outstanding. • Uniqueness and Factorization. M(μ,λ1,λ2) = ∫1 0 s λ1(1− s)λ2 dMμ(s)? This ques- tion can be rephrased in two equivalent ways. Do the positive integral moments of M(μ,λ1,λ2) as a function of μ and its Mellin transform asymptotic in the limit μ → 0 capture the distribution uniquely? Does ∫10 sλ1(1− s)λ2 dMμ(s) factorize as Lμ N(μ,λ1,λ2), where log Lμ is a zero-mean gaussian with vari- ance 2μ log 2 and N(μ,λ1,λ2) is an independent positive distribution having negative integral moments given as the coefficients of the power series in Corollary 3.2? • Arithmetic. Does M(q | μ, λ1, λ2) have an arithmetic interpretation? M(q | μ, λ1, λ2) has at least three properties that suggest a connection with ana- lytic number theory, which are its functional equations, and the occurrence of sums over Hurwitz zeta values in the “core” of its structure and of Bernoulli polynomials and Hurwitz and Riemann zeta values in its asymptotic expansion. • Inversion. What is the probability density function that corresponds to the Le´vy–Khinchine decompositions in Theorems 2.8 or 3.5? Equivalently, what is the Laplace transform inverse of the series in Corollaries 3.2 or 3.12? A natural first step is to compute the poles and residues of M(q | μ, λ1, λ2). Surprisingly, this calculation appears to depend on certain arith- metic properties of μ. at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4020 D. Ostrovsky • Transforms. Can Hardy’s moment constant method be generalized to sum the intermittency expansion in Corollary 4.6 for transforms other than the Mellin transform? Appendix In this section, we will give a brief review of the Alexeiewsky–Barnes function G(z | τ) defined in Equation (8). Our main references are Barnes’ original articles [10, 11]. This function is related to the double gamma function. Our goal is to remind the reader of this relationship, which implies the doubling formula in Equation (77), and then give proofs of Lemma 5.1, Lawrie and King’s formula, and Shintani’s identity. Our approach is ele- mentary and based on the fundamental characterization of the double gamma modular functions that was given by Barnes [10]. The main technical tool that is used throughout this section is the reduction of infinite sums over values of the log-gamma and digamma functions to explicit integrals by means of Malmste´n’s and Frullani’s formulas. We begin with the definition of the double gamma function that was introduced by Barnes in [11]. Assume �(ω1),�(ω2) > 0 for simplicity. Barnes succeeded in construct- ing a function Γ2(z | ω1, ω2) that is symmetric in ω1 and ω2 and satisfies the functional equations Γ −12 (z+ ω1 | ω1, ω2) Γ −12 (z | ω1, ω2) = ω z/ω2−1/2 2√ 2π Γ ( z ω2 ) , (A.1) Γ −12 (z+ ω2 | ω1, ω2) Γ −12 (z | ω1, ω2) = ω z/ω1−1/2 1√ 2π Γ ( z ω1 ) , (A.2) and the normalization condition lim z→0 [zΓ2(z | ω1, ω2)]= 1. (A.3) In [11], he proved the following general multiplication formula: mΓ2(mz | ω1, ω2) = e−2S0(mz) logm ∏m−1 r,s=0 Γ2(z+ (rω1 + sω2)/m)∏′ m−1 r,s=0 Γ2((rω1 + sω2)/m) , (A.4) 2S0(z)� z2 − z(ω1 + ω2) 2ω1ω2 . (A.5) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4021 Our interest in Γ2(z | ω1, ω2) has to do with its relationship to G(z | τ). Barnes [11] proved Γ −12 (z | ω1, ω2) = (2π)−z/2ω1ω1+2S0(z)2 G ( z ω1 ∣∣∣∣ω2ω1 ) . (A.6) Equivalently, we can write G(z | τ) = (2π)z/2τ−(1+2S0(z))Γ −12 (z | 1, τ ), (A.7) where 2S0(z) is now given as a function of τ by Equation (14). Hence, Γ2(z | ω1, ω2) is the symmetric version of G(z | τ), and the doubling formula in Equation (77) is a special case of Equation (A.4). Remark A.1. The Alexeiewsky–Barnes function is also a generalization of the better- known G-function of Barnes (cf. [45, Section 1.3] for a review), which arises in many different contexts. They are related by the formula G(z) =G(z | τ = 1). � We now proceed to compute what Barnes called the double gamma modular functions in the special case of �(τ ) > 0. Our computation is based on the following fun- damental result that he proved in [10]. Recall the definition of G(z | τ). Let Ω =mτ + n. G(z | τ)� exp ( a(τ ) z τ + b(τ ) z 2 2τ 2 ) z τ ∞∏ m,n=0 ′ ( 1+ z Ω ) exp ( − z Ω + z 2 2Ω2 ) . (A.8) Theorem A.2 (Barnes 1899). There exists double gamma modular functions C (τ ) and D(τ ) such that ∀z �= −τ,−2τ,−3τ, . . . ∞∏ m=1 [ 1 z+mτ e ψ(mτ)+(z+1/2)ψ ′(mτ) ] = Γ ( z τ + 1 ) τ z/τ+1/2√ 2π eC (τ )+(z+1/2)D(τ ). (A.9) The functions a(τ ) and b(τ ) satisfy the identities a(τ ) = τ 2 log(2πτ) + 1 2 log(τ ) − τC (τ ) + γ τ, (A.10) b(τ ) = −τ log(τ ) − τ 2D(τ ) − π 2τ 2 6 . (A.11) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4022 D. Ostrovsky Let a˜(τ )� (a(τ ) − γ )/τ , b˜(τ )� (b(τ ) + π2/6)/τ 2, then G(z+ 1 | τ) = ea˜(τ )z+b˜(τ )z2/2 ∞∏ m=0 Γ (1+mτ) Γ (z+ 1+mτ) e zψ(1+mτ)+z2ψ ′(1+mτ)/2. (A.12) � Using this characterization, we obtained the following result. Lemma A.3. Given �(τ ) > 0, a˜(τ ) = ∫∞ 0 dt t [ t (et − 1)(1− e−tτ ) − 1 etτ − 1 − e −tτ + e −tτ 2τ ] , (A.13) b˜(τ ) = ∫∞ 0 dt t [ e−tτ τ − t 2 (et − 1)(1− e−tτ ) ] . (A.14) � Proof. Recall Malmste´n’s formula and Frullani’s integral (cf. [47]). logΓ (1+ s) = ∫∞ 0 ( e−ts − 1 et − 1 + se −t ) dt t (Malmste´n), �(s) > −1, (A.15) log(s) = ∫∞ 0 (e−t − e−ts)dt t (Frullani), �(s) > 0. (A.16) Let |z| < |τ |. Applying Malmste´n’s formula to logΓ (z/τ + 1) and changing variables t′ = t/τ, we obtain logΓ ( 1+ z τ ) = ∫∞ 0 dt t ( e−tz − 1 etτ − 1 + z τ e−tτ ) . (A.17) Applying Malmste´n’s formula and Frullani’s integral to Equation (A.9), we obtain log ∞∏ m=1 [ 1 z+mτ e ψ(mτ)+(z+1/2)ψ ′(mτ) ] = ∫∞ 0 dt t 1 etτ − 1 ( e−tz + (z+ 1/2)t 2 − t 1− e−t ) . (A.18) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4023 Hence, log ∞∏ m=1 [ 1 z+mτ e ψ(mτ)+(z+1/2)ψ ′(mτ) ] − logΓ ( 1+ z τ ) = ∫∞ 0 dt t 1 etτ − 1 ( 1− t 1− e−t + 1 2 t2 1− e−t ) + z ∫∞ 0 dt t 1 etτ − 1 ( t2 1− e−t − 1 τ (1− e−tτ ) ) . (A.19) On the other hand, by Theorem A.2, this expression equals 1 2 log τ 2π + C (τ ) + 1 2 D(τ ) + z [ log τ τ + D(τ ) ] . (A.20) Equating the coefficients, we obtain C (τ ) + 1 2 log τ 2π = ∫∞ 0 dt t [ 1 etτ − 1 ( 1− t 1− e−t ) + 1 2 e−t τ ] , (A.21) D(τ ) = ∫∞ 0 dt t [ t2 (etτ − 1)(1− e−t) − e−t τ ] . (A.22) The result now follows by a straightforward algebraic reduction using Equations (A.10) and (A.11). � We now proceed to give elementary proofs of Lawrie and King’s formula, Lemma 5.1, and Shintani’s identity, based on Theorem A.2 and Lemma A.3. Proof of Lawrie and King’s formula, Equation (84). Recall Barnes’ formula for G(z | τ) in Equation (A.12). An application of Malmste´n’s formula to the infinite product in Equation (A.12) gives log ∞∏ m=0 Γ (1+mτ) Γ (z+ 1+mτ)e zψ(1+mτ)+z2ψ ′(1+mτ)/2 = ∫∞ 0 dt t [ 1− e−zt − zt+ z2t2/2 (et − 1)(1− e−tτ ) ] . (A.23) The result now follows by a straightforward algebraic reduction using the formulas for a˜(τ ) and b˜(τ ) that are given in Lemma A.3. � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4024 D. Ostrovsky Proof of Lemma 5.1. Let �(z) > −1. We start with Lawrie and King’s formula and group the terms in the integrand as follows: − logG(1+ z | τ) = ∫∞ 0 dt t [ 1 etτ − 1 [ e−zt − 1 et − 1 − (z 2 + z) t 2 + z ] + [ ze−tτ + e −zt − 1 et − 1 ] − (z 2 + z) 2τ 1 etτ − 1[1− e −tτ − tτ ] ] , = I (−z | 0, τ ) + logΓ (1+ z) − z log τ + (z 2 + z) 2τ γ, (A.24) where the last reduction follows by Malmste´n’s formula and Frullani’s integral. Hence, by Equation (11), − logG(1+ z+ τ | τ) = I (−z | 0, τ ) − (τ − 1) 2 log 2π + log τ 2 + (z 2 + z) 2τ γ, �(z+ τ) > −1. (A.25) On the other hand, using the definition of I (q | a, τ ) in Equation (78), I (q − a | 0, τ ) − I (−a | 0, τ ) = ∫∞ 0 dx x 1 exτ − 1 [ ex(q−a) − e−ax ex − 1 − q − (q2 − 2aq − q) 2 x ] . (A.26) It follows that we can write I (q | a, τ ) = I (q − a | 0, τ ) − I (−a | 0, τ ) − q ∫∞ 0 dx x 1 exτ − 1[e −ax − 1+ ax] − (q 2 − q) 2 ∫∞ 0 e−ax − 1 exτ − 1 dx. (A.27) Finally, it is easy to compute the remaining integrals. ∫∞ 0 dx x 1 exτ − 1[e −ax − 1+ ax]= logΓ (1+ a τ ) − a τ ψ(1), (A.28) ∫∞ 0 e−ax − 1 exτ − 1 dx= 1 τ [ ψ(1) − ψ ( 1+ a τ )] . (A.29) The result follows by combining Equations (A.25), (A.27)–(A.29). � at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from Selberg Integral as a Meromorphic Function 4025 We will end this section with an elementary proof of an identity that was dis- covered by Shintani [44] in 1980. Given z∈ C and |arg(τ )| < π, Γ2(z | 1, τ ) = (2π) z2 τ (z−z 2) 2τ − z2 eγ (z2−z) 2τ Γ (z) ∞∏ m=1 (mτ)1−ze (z−z2) 2mτ Γ (z+mτ) Γ (1+mτ) . (A.30) Proof of Shintani’s identity, Equation (75). It is sufficient to give the proof for �(z),�(τ ) > 0 and then extend the result to the specified domain by analytic continu- ation. Assume �(z), �(τ ) > 0. Recall the relationship between Γ2(z | 1, τ ) and G(z | τ) in Equation (A.7). Using the second functional equation of G(z | τ) it is easy to show that Equation (A.30) is equivalent to G(z+ τ | τ) = (2π) (τ−1)2 τ− 12 eγ (z−z 2) 2τ ∞∏ m=1 (mτ)z−1e (z2−z) 2mτ Γ (1+mτ) Γ (z+mτ) . (A.31) The key element in the proof is the following integral representation of the infinite prod- uct in Equation (A.31). log ∞∏ m=1 (mτ)z−1e (z2−z) 2mτ Γ (1+mτ) Γ (z+mτ) = γ (z 2 − z) 2τ + ∫∞ 0 dt t [ 1− e−t(z−1) (et − 1)(etτ − 1) − (z− 1) etτ − 1 + (z2 − z) 2τ e−tτ ] . (A.32) Using Malmste´n’s formula, Frullani’s integral, and the identity ∑M m=1 1/m= γ + logM + O(1/M), it is easy to see that we can write log ∞∏ m=1 (mτ)z−1e (z2−z) 2mτ Γ (1+mτ) Γ (z+mτ) = γ (z2 − z) 2τ + lim M→∞ [∫∞ 0 dt t [( 1− e−tMτ etτ − 1 )( 1− e−t(z−1) et − 1 − (z− 1) ) + (z 2 − z) 2τ (e−tτ − e−tMτ ) ]] . (A.33) Clearly, lim M→∞ [∫∞ 0 dt t [ − 1− e −t(z−1) (et − 1)(etτ − 1) + (z− 1) etτ − 1 − (z2 − z) 2τ ] e−tMτ ] = 0, (A.34) at U niversity of V irginia on D ecem ber 3, 2013 http://im rn.oxfordjournals.org/ D ow nloaded from 4026 D. Ostrovsky which gives us the identity in Equation (A.32). 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