Related papers: Confluent Vandermonde matrix and related topics
In this paper we outline a Matrix Ansatz approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We…
Orthogonal polynomials are of fundamental importance in many fields of mathematics and science, therefore the study of a particular family is always relevant. In this manuscript, we present a survey of some general results of the Hermite…
Based on Stokes' theorem we derive a non-holomorphic functional calculus for matrices, assuming sufficient smoothness near eigenvalues, corresponding to the size of related Jordan blocks. It is then applied to the complex conjugation…
A simple theory of the covariant derivatives, deformed derivatives and relative covariant derivatives of multivector and multiform fields is presented using algebraic and analytical tools developed in previous papers.
Using results of Fayers on the structure of Specht modules, we prove two different formulae for the determinant of matrices which are obtained by amalgamating the entries of two smaller matrices. In particular, this gives formulae for…
A nonstandard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered,…
A new and easy way of deriving Gauss's Generalized Hypergeometric Theorem is presented by using the Bilateral Binomial Theorem.
In this paper, by using the theory of circulant matrices we study some matrices over finite fields which involve the quadratic character and trinomial coefficients.
We give a concise direct proof of the orthogonality of interpolation Macdonald polynomials with respect to the Fourier pairing and briefly discuss some immediate applications of this orthogonality, such as the symmetry of the Fourier…
The paper introduces a method of partial fractions with matrix coefficients and its applications to finding chains of generalized eigenvectors, to evaluation of matrix exponentials, and to solution of linear systems of ordinary differential…
We generalise the Vandermonde determinant identity to one which tests whether a family of hypersurfaces in $\mathbf{P}^n$ has an unexpected intersection point.
We consider some combinatorial problems on matrix polynomials over finite fields. Using results from control theory we give a proof of a result of Helmke, Jordan and Lieb on the number of linear unimodular matrix polynomials over a finite…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
In 1989 we proposed to employ Vandermonde and Hankel multipliers to transform into each other the matrix structures of Toeplitz, Hankel, Vandermonde and Cauchy types as a means of extending any successful algorithm for the inversion of…
The Jacobian conjecture over a field of characteristic zero is considered directly in view of the nonlinear partial differential equations it is associated with. Exploring the integrals of such partial differential equations, this work…
Notions of the orthogonality and convolution orthogonality are explored with the use of the Kontorovich-Lebedev transform and its convolution. New classes of the corresponding orthogonal polynomials and functions are investigated. Integral…
Multipoint polynomial evaluation and interpolation are fundamental for modern symbolic and numerical computing. The known algorithms solve both problems over any field of constants in nearly linear arithmetic time, but the cost grows to…
The notion of Bezout matrix is an essential tool in studying broad variety of subjects: zeroes of polynomials, stability of differential equations, rational transformations of algebraic curves, systems of commuting nonselfadjoint operators,…
The second author recently suggested to identify the generating matrices of a digital $(t,m,s)$-net over the finite field $F_q$ with an $s \times m$ matrix $C$ over $F_{q^m}$. More exactly, the entries of $C$ are determined by interpreting…
We consider the problem of uniform interpolation of functions with values in a complex inner product space of finite dimension. This problem can be casted within a modified weighted pluripotential theoretic framework. Indeed, in the…