We compute correlation functions of inverse powers and ratios of characteristic polynomials for random matrix models with complex eigenvalues. Compact expressions are given in terms of orthogonal polynomials in the complex plane as well as their Cauchy transforms, generalizing previous expressions for real eigenvalues. We restrict ourselves to ratios of characteristic polynomials over their com...

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of p(A) over all monic polynomials p(z) of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well known propertie...

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called p, q -Fibonacci polynomials. We obtain combinatorial identities and by using Riordanmethodwe get factorizations of Pascal matrix involvin...

For random matrix ensembles with non-gaussian matrix elements that may exhibit some correlations, it is shown that centered traces of polynomials in the matrix converge in distribution to a Gaussian process whose covariance matrix is diagonal in the basis of Chebyshev polynomials. The proof is combinatorial and adapts Wigner’s argument showing the convergence of the density of states to the sem...

Abstract. The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix p...

The main aim of this paper is to construct a multivariable extension with the help of the extended Jacobi matrix polynomials (EJMPs). Generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived. Furthermore, general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented.

We present a positivity conjecture for the coefficients of the development of Jack polynomials in terms of power sums. This extends Stanley’s ex-conjecture about normalized characters of the symmetric group. We prove this conjecture for partitions having a rectangular shape.

This paper deals with a result concerning the algebraic dual of the linear mapping defined by the multiplication of polynomial vectors by a given polynomial matrix over a commutative field

We solve the problem of finding an enclosure for the range of a multivariate polynomial over a rectangular region by expanding the given polynomial into Bernstein polynomials. Then the coefficients of the expansion provide lower and upper bounds for the range and these bounds converge monotonically if the degree of the Bernstein polynomials is elevated. To obtain a faster improvement of the bou...

Jack function theory is useful for the calculation of the moment of the characteristic polynomials in Dyson’s circular β-ensembles (CβE). We define a q-analogue of the CβE and calculate moments of characteristic polynomials via Macdonald function theory. By this q-deformation, the asymptotics calculation of these moments becomes simple and the ordinary CβE case is recovered as q → 1. Further, b...