Related papers: Mellin transforms of multivariate rational functio…
We describe the complex of solutions of the algebraic Mellin transform of a $\mathcal{D}$-module $\mathcal{M}$ in terms of the solutions of $\mathcal{M}$. In order to do that, we define a Mellin functor on sheaves. We show the Mellin…
We define two pairings relating the A-motive with the dual A-motive of an abelian Anderson A-module. We show that specializations of these pairings give the exponential and logarithm functions of this Anderson A-module, and we use these…
In this note we show that for an arbitrary semisimple Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles.
We determine, up to exponentiating, the polar locus of the multivariable archimedean zeta function associated to a finite collection of polynomials F. The result is the monodromy support locus of F, a topological invariant. We give a…
We give a representation of the classical Riemann $\zeta$-function in the half plane $\Re s>0$ in terms of a Mellin transform involving the real part of the dilogarithm function with an argument on the unit circle (associated Clausen…
We present explicit representations in terms of hypergeometric functions for the scaling functions in the $C^0$ orthogonal multiresolution analyses associated with piecewise continuous polynomials. Closed formulas for the Mellin transform…
Towards the Hilbert-P\'olya conjecture, in this paper, we present a general construction of Hamiltonian $\hat{H}_{f},$ which leads a general family of Hurwitz zeta functions $(-1)^{z_{n}-1}L(f,z_{n},x+1)$ defined by Mellin transform becomes…
It is shown that Mellin transforms of p-adic Whittaker functions exist for generic characters. For good choices of vectors they are rational functions. For class one vectors they can be calculated explicitly. It turns out that they are…
We consider the Mellin transforms of certain generalized Hermite functions based upon certain generalized Hermite polynomials, characterized by a parameter $\mu>-1/2$. We show that the transforms have polynomial factors whose zeros lie all…
A simple scheme to express the Mellin transform of $D$-dimensional Euclidean conformal bootstrap equation is presented by relating conformal blocks to a Gauss-Grassmann (GG) system due to Gelfand-Graev, associated to conformal integrals,…
We propose to associate to a modular form (an infinite number of) complex valued functions on the $p$-adic numbers $\mathbb{Q}_p$ for each prime $p$. We elaborate on the correspondence and study its consequence in terms of the Mellin…
We investigate the degree of the polar transformations associated to a certain class of multi-valued homogeneous functions. In particular we prove that the degree of the pre-image of generic linear spaces by a polar transformation…
In this paper, Part II, of a two part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of $n\times n$ complex…
Let L(s) = L(s, \pi) be the standard L-function of a cuspidal representation \pi of GL(m,A) where A denotes the ad\`eles of the field of rationals. We consider the integral, on the real line Re(s)= 1/2, of the squared absolute value of…
We investigate the Mellin transforms of \(1/\operatorname{arctanh} x\) and \(1/(\sqrt{1-x^{2}}\,\operatorname{arctanh} x)\), viewed as compactly supported functions on \((0,1)\). These transforms are closely connected with conjectures on…
The article is devoted to holomorphic and meromorphic functions of quaternion and octonion variables. New classes of quasi-conformal and quasi-meromorphic mappings are defined and investigated. Properties of such functions such as their…
In the paper we consider a realization of a finite dimensional irreducible representation of the Lie algebra $\mathfrak{gl}_n$ in the space of functions on the group $GL_n$. It is proved that functions corresponding to Gelfand-Tsetlin…
We consider the algebras $M_p$ of Fourier multipliers and show that every bounded continuous function $f$ on $\mathbb R^d$ can be transformed by an appropriate homeomorphic change of variable into a function that belongs to $M_p(\mathbb…
The Mellin transform of fibre integral is calculated for certain isolated singularities of quasihomogeneous complete intersections (especially the unimodal singualrities of the list by Giusti and Wall). We show the property of symmetry…
Let $f$ be a rational map of degree $d\geq 2$. The moduli space $\mathcal{M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Latt\`es,…