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We survey general properties of multiplicative arithmetic functions of several variables and related convolutions, including the Dirichlet convolution and the unitary convolution. We introduce and investigate a new convolution, called gcd…
We construct the general permutation invariant Gaussian 2-matrix model for matrices of arbitrary size $D$. The parameters of the model are given in terms of variables defined using the representation theory of the symmetric group $S_D$. A…
To prove that a measure, linearly representable by means of a finite set of nonnegative matrices $\mathcal M$, has the weak-Gibbs property, one check the uniform convergence (on $\mathcal M^\mathbb N$) of the sequence of vectors…
The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional…
We present a unified integral framework based on the Fourier-Laplace transform evaluated along a vertical line in the complex plane. By identifying the moment-generating function (MGF) of a random variable with the weights of these…
We show that many important convex matrix functions can be represented as the partial infimal projection of the generalized matrix fractional (GMF) and a relatively simple convex function. This representation provides conditions under which…
In the paper, the authors introduce a matrix-parametrized generalization of the multinomial probability mass function that involves a ratio of several multivariate gamma functions. They show the logarithmic complete monotonicity of this…
In this article, we present a new two-dimensional generalization of the gamma function based on the product of the one-dimensional generalized beta function and the one-dimensional generalized gamma function. As will become clear later,…
According to generalized Mellin derivative (Kargin), we introduce a new family of polynomials called higher order generalized geometric polynomials. We obtain some properties of them.We discuss their connections to degenerate Bernoulli and…
Using the theory of $(\varphi, \Gamma)$-modules we generalizes Greenberg's construction of the $\Cal L$-invariant to semistable representations
We define generalised zeta functions associated to indefinite quadratic forms of signature (g-1,1) -- and more generally, to complex symmetric matrices whose imaginary part has signature (g-1,1) -- and we investigate their properties. These…
We discuss the product of $M$ rectangular random matrices with independent Gaussian entries, which have several applications including wireless telecommunication and econophysics. For complex matrices an explicit expression for the joint…
Generalized digamma functions $\psi_k(x)$, studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the zeta function associated to an indefinite quadratic form. In this paper, a…
We show the modular properties of the multiple 'elliptic' gamma functions, which are an extension of those of the theta function and the elliptic gamma function. The modular property of the theta function is known as Jacobi's…
We show in this paper that after proper scalings, the characteristic polynomial of a random unitary matrix converges almost surely to a random analytic function whose zeros, which are on the real line, form a determinantal point process…
Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…
Fractional moments have been investigated by many authors to represent the density of univariate and bivariate random variables in different contexts. Fractional moments are indeed important when the density of the random variable has…
New integral formulas involving the Meijer $G$-function are derived using recent results concerning distributional characterisations and distributional transformations in probability theory.
Our results can be viewed as applications of algebraic combinatorics in random matrix theory. These applications are motivated by the predictive power of random matrix theory for the statistical behavior of the celebrated Riemann…
Main purpose of this paper is to reconstruct generating function of the Bernstein type polynomials. Some properties this generating functions are given. By applying this generating function, not only derivative of these polynomials but also…