Related papers: Systems of orthogonal polynomials defined by hyper…
Orthogonal polynomials on quadratic curves in the plane are studied. These include orthogonal polynomials on ellipses, parabolas, hyperbolas, and two lines. For an integral with respect to an appropriate weight function defined on any…
Generalising the concept of a complete permutation polynomial over a finite field, we define completness to level $k$ for $k\ge1$ in fields of odd characteristic. We construct two families of polynomials that satisfy the condition of high…
The strict relation between some class of multiboson hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is used effectively to integrate the systems. As an explicit example we…
Orthogonal polynomials for the multivariate hypergeometric distribution are defined on lattices in polyhedral domains in $\RR^d$. Their structures are studied through a detailed analysis of classical Hahn polynomials with negative integer…
Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by…
We introduce the general polynomial algebras characterizing a class of higher order superintegrable systems that separate in Cartesian coordinates. The construction relies on underlying polynomial Heisenberg algebras and their defining…
In this paper, we derive a "hamiltonian formalism" for a wide class of mechanical systems, including classical hamiltonian systems, nonholonomic systems, some classes of servomechanism... This construction strongly relies in the geometry…
Orthogonal polynomials of two real variables can often be represented in complex variables. We explore the connection between the two types of representations and study the structural relations of complex orthogonal polynomials. The complex…
Orthogonal polynomials and multiple orthogonal polynomials are interesting special functions because there is a beautiful theory for them, with many examples and useful applications in mathematical physics, numerical analysis, statistics…
We characterize, up to a conjecture, probability distributions of all order finite moments having ultraspherical type generating functions for orthogonal polynomials.
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
Finite trigonometric Fourier series on a set of discrete equidistant points are considered. A finite system of orthogonal functions that have interpolation and certain differential properties on the period is introduced. Finite Fourier…
Many product formulas are known classically for generalized hypergeometric functions over the complex numbers. In this paper, we establish some analogous formulas for generalized hypergeometric functions over finite fields.
Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex…
We derive raising and lowering operators for orthogonal polynomials on the unit circle and find second order differential and $q$-difference equations for these polynomials. A general functional equation is found which allows one to relate…
We introduce sequences of functions orthogonal on a finite interval: proper orthogonal rational functions, orthogonal exponential functions, orthogonal logarithmic functions, and transmuted orthogonal polynomials
We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of…
Some particular examples of classical and quantum systems on the lattice are solved with the help of orthogonal polynomials and its connection to continuous models are explored.
Bilinear systems of equations are defined, motivated and analyzed for solvability. Elementary structure is mentioned and it is shown that all solutions may be obtained as rank one completions of a linear matrix polynomial derived from…
We generalize the known constructions of A-hypergeometric functions. In particular, we show that periods of middle dimension on affine or projective complex algebraic varieties are A-hypergeometric functions of coefficients of polynomial…