Biorthogonality of the Lagrange interpolants

We show that the Lagrange interpolation polynomials are biorthogonal with respect to a set of rational functions whose poles coinicde with interpolation point

Abstract "hypergeometric" orthogonal polynomials

We find all polynomials solutions $P_n(x)$ of the abstract "hypergeometric" equation $L P_n(x) = \lambda_n P_n(x)$, where $L$ is a linear operator sending any polynomial of degree $n$ to a polynomial of the same degree with the property that $L$ is two-diagonal in the monomial basis, i.e. $L x^n = \lambda_n x^n + \mu_n x^{n-1}$ with arbitrary nonzero coefficients $\lambda_n, \mu_n$ . Under obvious nondegenerate conditions, the polynomial eigensolutions $L P_n(x) = \lambda_n P_n(x)$ are unique. The main result of the paper is a classification of all {\it orthogonal} polynomials $P_n(x)$ of such type, i.e. $P_n(x)$ are assumed to be orthogonal with respect to a nondegenerate linear functional $\sigma$. We show that the only solutions are: Jacobi, Laguerre (correspondingly little $q$-Jacobi and little $q$-Laguerre and other special and degenerate cases), Bessel and little -1 Jacobi polynomials.Comment: 20 page

A Bochner Theorem for Dunkl Polynomials

We establish an analogue of the Bochner theorem for first order operators of Dunkl type, that is we classify all such operators having polynomial solutions. Under natural conditions it is seen that the only families of orthogonal polynomials in this category are limits of little and big $q$-Jacobi polynomials as $q=-1$

Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle

Using the technique of the elliptic Frobenius determinant, we construct new elliptic solutions of the $QD$-algorithm. These solutions can be interpreted as elliptic solutions of the discrete-time Toda chain as well. As a by-product, we obtain new explicit orthogonal and biorthogonal polynomials in terms of the elliptic hypergeometric function ${_3}E_2(z)$. Their recurrence coefficients are expressed in terms of the elliptic functions. In the degenerate case we obtain the Krall-Jacobi polynomials and their biorthogonal analogs

Elliptic Biorthogonal Polynomials Connected with Hermite's Continued Fraction

We study a family of the Laurent biorthogonal polynomials arising from the Hermite continued fraction for a ratio of two complete elliptic integrals. Recurrence coefficients, explicit expression and the weight function for these polynomials are obtained. We construct also a new explicit example of the Szeg\"o polynomials orthogonal on the unit circle. Relations with associated Legendre polynomials are considered.Comment: This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On the Polynomials Orthogonal on Regular Polygons

AbstractThe two-parameter Pastro–Al-Salam–Ismail (PASI) polynomials are known to be bi-orthogonal on the unit circle with continuous weight function when 0<q<1. We study the case ofqa root of unity. It is shown that corresponding PASI polynomials are orthogonal on the unit circle with discrete measure located on the vertices of the regularN-gon. Cases leading to a positive weight function are analyzed. In particular, we obtain trigonometric analogs of the Askey–Szego&#x030B; polynomials which are orthogonal on regularN-gons with positive weight function