On the Kampé de Fériet Hypergeometric Matrix Function

In this study, we derive recursion formulas for the Kampé de Fériet hypergeometric matrix function. We also obtain some finite matrix and infinite matrix summation formulas for the Kampé de Fériet hypergeometric matrix function.

FUZZY MODELS OF ANTAGONISTIC GAMES

The article is a continuation of a series of works on modeling situations in competitive markets at both micro and macro levels and the development of approaches to finding solutions to the obtained models. The paper proposes a method for solving a certain class of game-theoretic models under conditions of uncertainty. It is substantiated that a significant part of the problems of economic competition can be reduced to a finite matrix game of two players with zero sum, the matrix of winnings of the first player which has a specific form. Given the high degree of uncertainty in modern domestic markets and the need to simplify the current situation in its modeling due to the impossibility of including in the developed model of all real multifaceted relationships, the article considers antagonistic games with fuzzy parameters. It is proposed to look for the solution of the considered class of finite matrix games by reducing them to two dual optimization problems of linear programming with flexible limit constraints. The case is considered when the coefficients in the system of constraints of these models of linear programming are approximated by piecewise-linear membership functions, because they do not raise the question of linearity of the studied models. Using certain linear transformations, the optimization models of linear programming obtained in this work are reduced to models of a special kind, the method of solving which has been developed by other scientists. The essence of this method is that according to the Bellman-Zadeh approach, the resulting fuzzy model is reduced to the decision problem described by the multi-purpose optimization model, the solution of which includes only those alternatives, in such problems are called Pareto effective. Using this method, the fuzzy model obtained in the work is reduced to a "clear" problem of linear programming, some parameters of which are rationally determined by the person making managerial decisions, based on certain limitations obtained by solving two "clear" optimization models with known coefficients. By finding the solution to these dual problems and calculating the mixed strategies of the two players, the person making management decisions will be able to make the right choice among a set of alternative solutions.

Additive matrix convolutions of Pólya ensembles and polynomial ensembles

Recently, subclasses of polynomial ensembles for additive and multiplicative matrix convolutions were identified which were called Pólya ensembles (or polynomial ensembles of derivative type). Those ensembles are closed under the respective convolutions and, thus, build a semi-group when adding by hand a unit element. They even have a semi-group action on the polynomial ensembles. Moreover, in several works transformations of the bi-orthogonal functions and kernels of a given polynomial ensemble were derived when performing an additive or multiplicative matrix convolution with particular Pólya ensembles. For the multiplicative matrix convolution on the complex square matrices the transformations were even done for general Pólya ensembles. In the present work, we generalize these results to the additive convolution on Hermitian matrices, on Hermitian anti-symmetric matrices, on Hermitian anti-self-dual matrices and on rectangular complex matrices. For this purpose, we derive the bi-orthogonal functions and the corresponding kernel for a general Pólya ensemble which was not done before. With the help of these results, we find transformation formulas for the convolution with a fixed matrix or a random matrix drawn from a general polynomial ensemble. As an example, we consider Pólya ensembles with an associated weight which is a Pólya frequency function of infinite order. But we also explicitly evaluate the Gaussian unitary ensemble as well as the complex Laguerre (aka Wishart, Ginibre or chiral Gaussian unitary) ensemble. All results hold for finite matrix dimension. Furthermore, we derive a recursive relation between Toeplitz determinants which appears as a by-product of our results.