We provide the mathematical foundation for the Xm-Jacobi spectral theory. Namely, we present a self-adjoint operator associated to the differential expression with the exceptional Xm-Jacobi orthogonal polynomials as eigenfunctions. This proves that those polynomials are indeed eigenfunctions of the self-adjoint operator (rather than just formal eigenfunctions). Further, we prove the completenes...

Let be a closed oriented contour on the Riemann sphere. Let E and F be polynomials of degree n + 1, with zeros respectively on the positive and negative sides of . We compute the [n=n] and [n 1=n] Padé denominator at 1 to f (z) = Z 1 z t dt E (t)F (t) : As a consequence, we compute the nth orthogonal polynomial for the weight 1= (EF ). In particular, when is the unit circle, this leads to an ex...

We study nonsymmetric second order diierence operators acting in the Hilbert spacè 2 and describe the resolvent set and the essential spectrum of such operators in terms of related formal orthogonal polynomials. As an application, we obtain new results on the growth of orthonormal polynomials outside and inside the support of the underlying measure of orthogonality.

We consider the problem of computing all the zeros of an analytic function that lie in the interior of a Jordan curve, together with their respective multiplicities. Our approach uses modiied moments based on formal orthogonal polynomials. Numerical experiments indicate that it is far superior to classical approaches, which consider the usually ill-conditioned map from the Newton sums to the un...

Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix r new rows and columns, so that the original Jacobi matrix is shifted downward. The r new...

in this study a numerical method is developed to solve the hammerstein integral equations. to this end the kernel has been approximated using the leastsquares approximation schemes based on legender-bernstein basis. the legender polynomials are orthogonal and these properties improve the accuracy of the approximations. also the nonlinear unknown function has been approximated by using the berns...

in this study, the problem of determining an optimal trajectory of a nonlinear injection into orbit problem with minimum time was investigated. the method was based on orthogonal polynomial approximation. this method consists of reducing the optimal control problem to a system of algebraic equations by expanding the state and control vector as chebyshev or legendre polynomials with undetermined...

A recent result of S.-Y. Lee and M. Yang states that the planar orthogonal polynomials with respect to a modified Gaussian measure are multiple type II on contour in complex plane. We show same also I contour, provided exponents weight integer. From this orthogonality, we derive several equivalent Riemann-Hilbert problems. The proof is based fundamental identity Yang, which establish using new ...

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 asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic alg...