Related papers: A note on the Geronimus transformation and Sobolev…
We introduce a new multivariate orthogonal polynomial which is a 2-parameter deformation of the spherical polynomial by harmonic analysis on symmetric cone. This is also regarded as a multivariate analogue of the circular Jacobi polynomial.…
Given a parametrised weight function $\omega(x,\mu)$ such that the quotients of its consecutive moments are M\"obius maps, it is possible to express the underlying biorthogonal polynomials in a closed form \cite{IN2}. In the present paper…
In this paper we derive the bi-orthogonality relations, diagonal term evaluations and evaluation formulas for the non-symmetric Koornwinder polynomials. For the derivation we use certain representations of the (double) affine Hecke algebra…
We prove local and global invertibility of Sobolev solutions of certain differential inclusions which prevent the differential matrix from having negative eigenvalues. Our results are new even for quasiregular mappings in two dimensions.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and…
We investigate several families of multiple orthogonal polynomials associated with weights for which the moment generating functions are hypergeometric series with slightly varying parameters. The weights are supported on the unit interval,…
In this paper, we study a class of orthogonal polynomials defined by a three-term recurrence relation with periodic coefficients. We derive explicit formulas for the generating function, the associated continued fraction, the orthogonality…
In this sequel to arxiv:arXiv:1012.0835 we develop Bezout type theorems for semidegrees (including an explicit formula for {\em iterated semidegrees}) and an inequality for subdegrees. In addition we prove (in case of surfaces) a Bernstein…
We survey various classical results on invariants of polynomials, or equivalently, of binary forms, focussing on explicit calculations for invariants of polynomials of degrees 2, 3, 4.
In this paper, we are interested in the construction of a bilinear pseudodifferential calculus. We define some symbolic classes which contains those of Coifman-Meyer. These new classes allow us to consider operators closely related to the…
In a previous paper (arXiv:1410.5207) certain birational transformations were constructed between the noncommutative schemes associated to quadratic and cubic three dimensional Sklyanin algebras. In the current paper we consider the inverse…
We evaluate the hyperpfaffian of a skew-symmetric $k$-ary polynomial $f$ of degree $k/2 \cdot (n-1)$. The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial $f$. The proof…
The Witt ring of symmetric bilinear forms over a field has divided power operations. On the other hand, it follows from Garibaldi-Merkurjev-Serre's work on cohomological invariants that all operations on the Witt ring are essentially linear…
The purpose of this paper is to obtain Fourier transforms of multivariate orthogonal polynomials on the cone such as Laguerre polynomials on the cone and Jacobi polynomials on the cone and to define two new families of multivariate…
We survey our recently proposed method for constructing biholomorphic invariants of quasihomogeneous isolated hypersurface singularities and, more generally, invariants of graded Artinian Gorenstein algebras. The method utilizes certain…
A generalized derivative nonlinear Schr\"odinger equation, \ii q_t + q_{xx} + 2\ii \gamma |q|^2 q_x + 2\ii (\gamma-1)q^2 q^*_x + (\gamma-1)(\gamma-2)|q|^4 q = 0 , is studied by means of Hirota's bilinear formalism. Soliton solutions are…
We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with…
We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence…
By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms…
Two unitary integral transforms with a very-well poised $_7F_6$-function as a kernel are given. For both integral transforms the inverse is the same as the original transform after an involution on the parameters. The $_7F_6$-function…