Selberg Integral as a Meromorphic Function

The Selberg integral (the integral of the discriminant on n variables raised to a power −μ/2 times the beta prefactor $\Pi_i\ x^{\lambda_1}_i(1-x_i)^{\lambda_2}$ over the unit n-interval) is analytically extended as a function of its dimension n to the complex plane. The resulting meromorphic function $\mathfrak M(q|\mu,\lambda_1,\lambda_2)$ 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 of $\mathfrak M(q|\mu,\lambda_1,\lambda_2)$ is shown to coincide with the intermittency expansion of the Mellin transform of a particular functional of the limit lognormal stochastic process resulting in the conjecture that $\mathfrak M(q|\mu,\lambda_1,\lambda_2)$ 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.