On Boundary Values of Hypergeometric Functions of 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 the Euler integral representations of hypergeometric functions in several variables

Fractional integration of the H-function of several variables

This theorem of F. Riesz' has the distinction of being true only for analytic functions and not for harmonic functions in general as many similar theorems are. Correspondingly, the proof is very analytic. It is based on the canonical decomposition f(z) = g(z).B(z) where g(z) $ 0 and B(z) has boundary values 1 almost everywhere (Blaschke product). Now, for functions of several complex variables no comparable decomposition exists, and yet we will succeed in generalizing the theorem of Riesz proper. If f(z, ***,z) is analytic for O < I zj I < 1,j = 1, ... k, and if

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

Elements of[formula omitted]′Ls(Mp)and[formula omitted]′Ls{Mp}as Boundary Values of Holomorphic Functions

The purpose of this paper is to analyze the accuracy of calculations and the amount of time spent on finding optimal values for functions of several variables using the algorithmic stochastic optimization algorithms based on the annealing simulation method. To conduct research by the staff of the Department of General Mechanics of the Lipetsk State Technical University, software has been created that implements algorithms for finding extremal values for functions of several variables. The functional purpose of the software is intellectual support for decision-making in the formation of chemical compositions of cast iron alloys. Optimization is performed on a specific and fixed search area, which is a hyperparallelepiped, the boundaries of the variation in the percentage content of a chemical element in the alloy. In the program, ten modifications of the annealing simulation algorithm are realized, allowing a finite number of steps to make an estimate of the optimal value of the input elements of the function under study on a multidimensional space. In particular, modification of A, B and С algorithm schemes using the Boltzmann and Cauchy distribution functions, as well as the superfast annealing algorithm and the Xin Yao algorithm are implemented. The results of the computational experiment for three different functions are presented: from six, eight and nine variables, a comparative analysis of the proximity of the results to the optimal values for each modification of the algorithm is carried out. The obtained data made it possible to draw conclusions about the advantages and disadvantages of each of the modifications of the stochastic search algorithm. Based on the results of computational experiments, the regularities between the time costs of algorithms and their numerical parameters are determined. With the help of linear regression, analytical dependencies are obtained, which allow estimating the time of the algorithms operation depending on the values of their numerical parameters. The paper contains a number of conclusions about the applicability of stochastic optimization modifications for solving applied problems, the shortcomings of modifications and their dignity are highlighted.

We derive a set of integral equations which are necessary and sufficient conditions on the form factors of local field theory, i.e. on the matrix elements of local operators. The basic idea is that out of the set of all (distribution-valued) functions defined on the boundary of an analyticity domain, in general only a subset are boundary values of functions which are analytic within the domain. The form factors are boundary values of a vertex function which, due to the general assumptions of locality, reasonable energy, and mass spectrum and Poincaré covariance, is analytic at least in the domain constructed by Källén and Wightman. The characteristic boundary of the domain (``the distinguished boundary'') is the set of physical values of the arguments of the form factors, and the integral equations in that way only involve such values. The main advantages in formulating the locality conditions in this way are that (1) only the physical quantities of the field theory, i.e., the matrix elements between the field operators, enter into the equations and (2) the frustrating complications which are met in the construction of the domains of analyticity for n-point functions with n &amp;gt; 3 might hopefully be avoided because the distinguished boundary can be constructed even if the whole domain is not known. The integral equations have naturally no unique solutions, because, e.g., all perturbation-theoretical form factors must, of course, fulfill them. The equations may, however, function as a convenient starting point for approximations and ``model building'' for form factors outside the presently used perturbation theories. The integral equations are straightforward generalizations of the notion of ``weak local commutativity'' for the two-point function. This condition means that the two spectral functions connected to two locally commuting operators should be equal. The conditions on the form factors (which are the generalizations of the two-point spectral functions) are that the difference should vanish when integrated over certain physical sets of mass space.

It is shown that if n ∈ ℕ, c ∈ ℂn, and three distinct values of a meromorphic function f: ℂn sr 1 of hyper-order gV(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂn sr ℂn, τ (z) = z + c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green’s Picard-Type Theorem in the special case where gV(f) < 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions ℂn → ℙ P1 are given, and their applications to partial difference equations are discussed.

Let 𝔽q be a finite field with q = pe elements, where p is a prime. We calculate the number of zeros of the hyperelliptic curve by using the properties of the quadratic character. With the help of these zeros we discuss the trace of Frobenius map on the hyperelliptic curves in terms of special values of 2F1 hypergeometric functions. Also, we find some identities, and particular values of hypergeometric functions over 𝔽q.

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.

Computation of the largest value of functions of several variables

0. A classical result of Fatou states that if u(z) is harmonic and bounded in lzl < 1 then u has a nontangential limit at almost every point ei?. The same conclusion holds if u is only bounded from below. These results have a local analogue -namely, if u(z) is harmonic in lzl < 1 and at each point ei0 of a measurable set E there is some cone in which u(z) is bounded from either above or below then u(z) has a nontangential limit at almost every eiO & E. With the aid of conformal mapping one can show the above results hold for regions of the plane more general than lzl < 1. Methods similar to those used for lzl < 1 may be applied to functions which are harmonic in the unit ball of Euclidean (n + 1)-space. However, since we lack a conformal mapping theorem for more general domains DcE +1 the situation there is more technically complicated. Known results for certain kinds of domains D c E, + 1 are due to Brelot and Doob [2], Calderon [3], Carleson [4] and Widman [7], and the purpose of this paper is to obtain nontangential boundary values for functions which are harmonic in still more general types of domains D' En + 1 and bounded from above or below in cones. The domains which we consider are all regular domains for the solution of the Dirichlet problem. For a regular domain D, our result may be stated as follows. Let EC aD and suppose for each Q E E there is a cone with vertex Q which is exterior to D. Then if u is harmonic in D and is bounded from above or below in a cone with vertex Q for each Q E E, u has a finite nontangential limit at each Q E E except for a set of harmonic measure zero. It is natural that the exceptional set be one of harmonic measure zero since if Ec AD is any set of harmonic measure zero it is easy to construct a positive harmonic function with boundary value +oo at each point of E. Our main result is a consequence of the basic result that if u is harmonic and bounded in a starlike Lipschitz domain D, then u has a finite nontangential limit at each Q E AD except for a set of harmonic measure zero. In ?1 of this paper we define the terms starlike Lipschitz domain, nontangential limit, and harmonic measure. We also include some elementary consequences of these definitions and some essential theorems on the differentiation of integrals.

Denote by \(H_{\pm}\) the upper respectively lower half plane of the complex plane \(\mathbb C\). For functions \(F_{\pm}\) which are analytic in \(H_{\pm}\) we introduce the concept of boundary value in the sense of distributions and show under which conditions (growth restrictions near the boundary) such an analytic function has a distributional boundary value. For distributions T with compact support \(K \subset \mathbb{R}\) it is straightforward to show that there is a function F analytic in \(\mathbb C \backslash K\) such that T is the difference of boundary values \(F(x +\,\textrm{i}o) - F(x -\,\textrm{i}o)\) of F from \(H_{\pm}\). Without proof we state the result that every distribution T on the real line is the difference of the boundary values of two functions \(F_{\pm}\) analytic in \(H_{\pm}\): \(T(x)= F_+(x +\,\textrm{i}o)-F_-(x-\,\textrm{i}o)\). This result can be extended to the case of distributions in more than one variable. We indicate briefly the main difficulties and the solution by Martineau.

Holomorphic functions in tubes which generalize the Hardy Hp functions are considered. In previous work the existence of strong l′ distributional boundary values for the holomorphic functions has been established. In this paper we show that if these boundary values are in the distribution space then the holomorphic functions can be recovered by the Cauchy and Poisson integrals of the boundary values.