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For any positive integer $N$, we describe a natural complex representation of the symmetric group $\Sigma_N$ on the vector space spanned by its involutions that contains each irreducible representation exactly once.
We obtain a family of matrix integrals which decompose to a product of Gamma-functions (they have some relations with S.G.Gindikin 'Beta', but generally speaking essentially differ from it). We obtain Plancherel formula for Berezin…
This paper presents a family of new integral representations and asymptotic series of the multiple gamma function. The numerical schemes for high-precision computation of the Barnes gamma function and Glaisher's constant are also discussed.
A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this…
The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new…
For any discrete group $\Gamma$ and any 2-dimensional complex representation $\rho$ of $\Gamma$, we introduce the notion of $\rho-$equivariant functions, and we show that they are parameterized by vector-valued modular forms. We also…
A generalized matrix function is a generalization of determinant and permanent function. In this paper, we introduced the formula for the value of a generalized matrix function of a linear sum of permutation matrices. We show that a linear…
In this paper, the deformation of the Heisenberg algebra, consistent with both the generalized uncertainty principle and doubly special relativity, has been analyzed. It has been observed that, though this algebra can give rise to…
We show how the formulas in paper Variae considerationes circa series hypergeometricas written by Euler imply the duplication formula for the Gamma-function. This paper can be seen as an Addendum to a previous paper by the author.
We investigated an integral representation of a complex function which was obtained from the molecular beam magnetic resonance, and named it as `$g$- function' because it was connected with Gamma function as a special case. We introduced a…
In the present article the reduced integral representation of partitions in terms of harmonic products has been derived first by using hypergeometry and the new concept of fractional sum and secondly by studying the Fourier series of the…
We introduce a simple procedure to integrate differential forms with arbitrary holomorphic poles on Riemann surfaces. It gives rise to an intrinsic regularization of such singular integrals in terms of the underlying conformal geometry.…
In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where…
Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function $\eta(s)$, and hence Riemann's function $\zeta(s)$, is obtained in terms of the Exponential Integral function $E_{s}(i\kappa)$ of…
A Green's function approach is presented for the D-dimensional inverse square potential in quantum mechanics. This approach is implemented by the introduction of hyperspherical coordinates and the use of a real-space regulator in the…
In this paper we deal with Mellin convolution of generalized Gamma densities which leads to integrals of modified Bessel functions of the second kind. Such convolutions allow us to explicitly write the solutions of the time-fractional…
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,…
Spherical Bessel functions appear commonly in many areas of physics wherein there is both translation and rotation invariance, and often integrals over products of several arise. Thus, analytic evaluation of such integrals with different…
A transformation on homogeneous polynomials is proposed, which is further applied to parametric Feynman integrals. The two representations related through this transformation are dual to each other. It naturally leads to dualities of Landau…
Two kinds of infinite product representations for Vign\'eras multiple gamma function are presented. As an application of these formulas, a multiplication formula for the function is derived.