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We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…
Multiplication of polynomials is among key operations in computer algebra which plays important roles in developing techniques for other commonly used polynomial operations such as division, evaluation/interpolation, and factorization. In…
We introduce a multivariate analogue of Bernoulli polynomials and give their fundamental properties: difference and differential relations, symmetry, explicit formula, inversion formula, multiplication theorem, and binomial type formula.…
We present a concentration inequality for linear functionals of noncommutative polynomials in random matrices. Our hypotheses cover most standard ensembles, including Gaussian matrices, matrices with independent uniformly bounded entries…
In this note we show that the degree of the interpolation polynomial for equidistant base points is characterized by the regularity of matrices of combinatorical type.
The purpose of this note is to extend in a simple and unified way the known results on interlacing of zeros of paraorthogonal polynomials on the unit circle. These polynomials can be regarded as the characteristic polynomials of any matrix…
The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation…
We consider the problem of complex root classification, i.e., finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots. It is well known that such conditions can be…
In this work we develop the theory of Gr\"obner bases for modules over the ring of univariate linearized polynomials with coefficients from a finite field.
A new characterization of the generalized Hermite polynomials and of the orthogonal polynomials with respect to the maesure $|x|^\g (1-x^2)^{\a-1/2}dx$ is derived which is based on a "reversing property" of the coefficients in the…
We derive inclusion regions for the eigenvalues of matrix polynomials expressed in a general polynomial basis, which can lead to significantly better results than traditional bounds. We present several applications to engineering problems.
We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct…
In this paper we introduce and discuss some classes of orthogonal polynomials in several non-commuting variables. The emphasis is on a non-commutative version of the orthogonal polynomials on the real line. We introduce recurrence equations…
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…
Differential properties for orthogonal polynomials in several variables are studied. We consider multivariate orthogonal polynomials whose gradients satisfy some quasi--orthogonality conditions. We obtain several characterizations for these…
A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…
We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Vandermonde…
The new method for obtaining a variety of extensions of Hermite polynomials is given. As a first example a family of orthogonal polynomial systems which includes the generalized Hermite polynomials is considered. Apparently, either these…
This paper considers the extension of classical Lagrange interpolation in one real or complex variable to "polynomials of one quaternionic variable". To do this we develop some aspects of the theory of such polynomials. We then give a…
We generalize the differential dimension polynomial from prime differential ideals to characterizable differential ideals. Its computation is algorithmic, its degree and leading coefficient remain differential birational invariants, and it…