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The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the…
We consider biorthogonal polynomials that arise in the study of a generalization of two--matrix Hermitian models with two polynomial potentials V_1(x), V_2(y) of any degree, with arbitrary complex coefficients. Finite consecutive…
This paper revisits the notion of classical orthogonal polynomials from a broader functional-analytic point of view. It is intended neither as a survey of known results nor as a review of the literature, but rather as a conceptual…
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
In this paper we study the subdifferential set of an operator. We give possible relation of the subdifferential set of an operator to that of its value, at a point where the operator attains its norm.
In this article, the 2-iterated Sheffer polynomials are introduced by means of generating function and operational representation. Using the theory of Riordan arrays and relations between the Sheffer sequences and Riordan arrays, a…
We show the existence of polynomial maps which have a regular bifurcation value, while over a neighbourhood of this value the fibres are connected and diffeomorphic.
The purpose of this paper is to find the characterization of the Sheffer polynomial sets satisfying the d-orthogonality conditions. The generating function form of these polynomial sets is given in Theorem 2.2. As applications of the…
In the present paper, we introduce a method to construct two variable biorthogonal polynomial families with the help of one variable biorthogonal and orthogonal polynomial families. By using this new technique, we define 2D Hermite…
We demonstrate in some detail how Macdonald polynomials emerge from the recently introduced 3-Schur functions when the plane-partition vector time-variables are projected onto the ordinary scalar times under non-vanishing angles, which…
A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…
In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004…
We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an…
We explore the relationship between polynomial functors and (rooted) trees. In the first part we use polynomial functors to derive a new convenient formalism for trees, and obtain a natural and conceptual construction of the category…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula…
A three-dimensional polynomial algebra of order $m$ is defined by the commutation relations $[P_0, P_\pm]$ $=$ $\pm P_\pm$, $[P_+, P_-]$ $=$ $\phi^{(m)}(P_0)$ where $\phi^{(m)}(P_0)$ is an $m$-th order polynomial in $P_0$ with the…
Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. Let be a two-dimensional timescale $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ such that $\mathbb{T}_1 =…
Multi-degree splines are smooth piecewise-polynomial functions where the pieces can have different degrees. We describe a simple algorithmic construction of a set of basis functions for the space of multi-degree splines, with similar…
An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…