On the Euler integral representations of hypergeometric functions in several variables

While investigating the Lauricella's list of 14 complete second-order hypergeometric series in three variables, Srivastava noticed the existence of three additional complete triple hypergeometric series of the second order, which were denoted by <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX>. Each of these three triple hypergeometric functions <TEX>$H_A$</TEX>, <TEX>$H_B$</TEX> and <TEX>$H_C$</TEX> has been investigated extensively in many different ways including, for example, in the problem of finding their integral representations of one kind or the other. Here, in this paper, we aim at presenting further integral representations for the Srivatava's triple hypergeometric function <TEX>$H_B$</TEX>.

On Boundary Values of Hypergeometric Functions of Several Variables

Fractional integration of the H-function of several variables

A functional system is a set of functions endowed with a set of operations on these functions. The operations allow one to obtain new functions from the existing ones.Functional systems are mathematical models of real and abstract control systems and thus are one of the main objects of discrete mathematics and mathematical cybernetic.The problems in the area of functional systems are extensive. One of the main problems is deciding completeness that consists in the description of all subsets of functions that are complete, i.e. generate the whole set.The well-known Kolmogorov theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition adjoins the completeness problem (see the formulation of this theorem below).The purpose of this paper is the following problem: is there an analogue of Kolmogorov’s theorem on the representation of continuous functions of several variables in the form of superpositions of continuous functions of one variable and addition for functional systems of polynomial functions and functional systems of rational functions?It turns out that the problem posed (an analogue of Kolmogorov’s theorem) has a negative answer for functional systems of polynomial functions with natural and integer coefficients, and for functional systems of polynomial functions with rational and real coefficients and for functional systems of rational functions with rational and real coefficients - the answer is positive. These theorems are the main results of this paper.

On the structure of representations of continuous functions of several variables as finite sums of continuous functions of one variable

On the representation of continuous functions of several variables by superpositions of continuous functions of a smaller number of variables

On the Structure of Representations of Continuous Functions of Several Variables as Finite Sums of Continuous Functions of One Variable

ABSTRACTWe introduce q-Lauricella matrix functions of three and n-variables. The convergence of these matrix functions is examined for both and cases. The systems of matrix q-difference equations satisfied by these q-Lauricella matrix functions as well as basic Srivastava's triple hypergeometric matrix functions have been determined. Certain integral representations satisfied by these matrix functions have also been established.

Dyadic calculus and sampling theorems for functions with multidimensional domain II. Applications to dyadic sampling representations

On Kolmogorov's representation of functions of several variables by functions of one variable

On the representation of functions of several variables as a superposition of functions of a smaller number of variables

The concept of solid texturing is extended in two directions: constructive modeling of space partitions for texturing and modeling of multi-dimensional textured objects called hypervolumes. A hypervolume is considered as a point set with attributes of both physical (density, temperature, etc.) and photometric (color, transparency, diffuse and specular reflections, etc.) nature. The point-set geometry and attributes are modeled independently using real-valued scalar functions of several variables. Each real-valued function defining the geometry or an attribute is evaluated in the given point by a procedure traversing a constructive tree structure with primitives in the leaves and operations in the nodes of the tree. This approach provides a framework for the modeling, texturing and visualization of 3D solids and time-dependent and multi-dimensional objects in a completely uniform manner. We introduced a special modeling language and implemented software tools supporting the proposed approach. The concept of constructive hypervolume textures is independent of the geometric representation. We provide examples of textured functional representation (F-Rep) and boundary representation (B-Rep) objects as illustrations.

Associated to any finite reflection group G on an Euclidean space there is a parametrized commutative algebra of differential-difference operators with as many parameters as there are conjugacy classes of reflections.The Hecke algebra of the group can be represented by monodromy action on the space of functions annihilated by each differential-difference operator in the algebra.For each irreducible representation of G the differential-difference equations lead to a linear system of first-order meromorphic differential equations corresponding to an integrable connection over the G-orbits of regular points in the complexification of the Euclidean space.The fundamental group is the generalized Artin braid group belonging to G, and its monodromy representation factors over the Hecke algebra of G. Monodromy has long been of importance in the study of special functions of several variables, for example, the hyperlogarithms of Lappo-Danilevsky are used to express the flat sections and the work of Riemann on the monodromy of the hypergeometric equation is applied to the case of dihedral groups.

4. A. Garsia, S. Orey and E. Rodemich, Asymptotic behavior of successive coefficients of some power series, Illinois J. Math. 6 (1962), 620-629. 5. S. Karlin and J. McGregor, Random walks, Illinois J. Math. 3 (1959), 66-81. 6. D. G. Kendall, Unitary dilations of Markov transition operators, and the corresponding integral representation for transition-probability matrices, Harold Cram6r Volume (U. Grenander, Ed.), Almquist and Wiksell, Stockholm, 1960; pp. 139-161. 7. S. Orey, Strong ratio limit property, Bull. Amer. Math. Soc. 67 (1961), 571-574. 8. D. Vere-Jones, Geometric ergodicity in denumerable Markov chains, Quart. J. Math. Oxford Ser. (2) 13 (1962), 7-28.

Integral representations play a prominent role in the analysis of entire functions. The representations of generalized Mittag-Leffler type functions and their asymptotics have been (and still are) investigated by plenty of authors in various conditions and cases. The present paper explores the integral representations of a special function extending to two variables the two-parametric Mittag-Leffler type function. Integral representations of this functions within different variation ranges of its arguments for certain values of the parameters are thus obtained. Asymptotic expansion formulas and asymptotic properties of this function are also established for large values of the variables. This yields corresponding theorems providing integral representations as well as expansion formulas.