We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, fea...

X iv :m at h/ 05 12 22 2v 1 [ m at h. SP ] 1 1 D ec 2 00 5 The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences Leonid Golinskii and Stefano Serra-Capizzano 1 Abstract Under the mild trace-norm assumptions, we show that the eigenvalues of a generic (non Hermitian) complex perturbation of a Jacobi matrix sequence (not necessarily real) are still distri...

In this paper we determine the complete coset weight distributions of the second order Reed-Muller code RM(2, 6) of length 64. Our method fully uses the interaction between the Jacobi polynomials for the code RM(2, 6) and those of the dual code RM(3, 6). The method also employs the group theoretic reduction processes to diminish the runtimes of computing the Jacobi polynomials for the code RM(2...

Polynomials whose coefficients are successive derivatives of a class of Jacobi polynomials evaluated at x = 1 are stable. This yields a novel and short proof of the known result that the Bessel polynomials are stable polynomials. Stability preserving linear operators are discussed. The paper concludes with three open problems involving the distribution of zeros of polynomials.

An explicit structure relation for Askey-Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey-Wilson inner product and which sends polynomials of degree n to polynomials of degree n+ 1. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtain...

We extend a collocation method for solving a nonlinear ordinary differential equation ODE via Jacobi polynomials. To date, researchers usually use Chebyshev or Legendre collocation method for solving problems in chemistry, physics, and so forth, see the works of Doha and Bhrawy 2006, Guo 2000, and Guo et al. 2002 . Choosing the optimal polynomial for solving every ODEs problem depends on many f...

In this paper, an application to the approximation by wavelets has been obtained by using matrix-Cesàro (Λ · C1) method of Jacobi polynomials. The rapid rate of convergence of matrix-Cesàro method of Jacobi polynomials are estimated. The result of Theorem (6.1) of this research paper is applicable for avoiding the Gibbs phenomenon in intermediate levels of wavelet approximations. There are majo...

We introduce two kinds of multiple little q-Jacobi polynomials p~n with multi-index ~n = (n1, n2, . . . , nr) and degree |~n| = n1 + n2 + · · · + nr by imposing orthogonality conditions with respect to r discrete little q-Jacobi measures on the exponential lattice {qk, k = 0, 1, 2, 3, . . .}, where 0 < q < 1. We show that these multiple little qJacobi polynomials have useful q-difference proper...

We study some properties of the Askey-Wilson polynomials (AWP) when q is a primitive Nth root of unity. For general four-parameter AWP, zeros of the Nth polynomial and the orthogonality measure are found explicitly. Special subclasses of the AWP, e.g., the continuous q-Jacobi and big q-Jacobi polynomials, are considered in detail. A set of discrete weight functions positive on a real interval i...

Abstract A joint algebraic interpretation of the biorthogonal Askey polynomials on unit circle and orthogonal Jacobi is offered. It ties their bispectral properties to an algebra called meta-Jacobi $m\mathfrak {J}$ .