We generalize the combinatorial interpretations of orthogonal polynomials found in [2] in two directions: first, we show that the interpretation of moments as a generating function of Motzkin paths may be generalized to no only different linear functionals, but to arbitrary sequences of polynomials. Second, we develop several new involutions to describe the L norm of polynomials that satisfy a ...

Given a symmetric Sobolev inner product of order N , the corresponding sequence of monic orthogonal polynomials {Qn} satisfies that Q2n(x) = Pn(x), Q2n+1(x) = xRn(x) for certain sequences of monic polynomials {Pn} and {Rn}. In this paper, we deduce the integral representation of the inner products such that {Pn} and {Rn} are the corresponding sequences of orthogonal polynomials. Moreover, we st...

Multivariate orthogonal polynomials can be introduced by using a moment functional defined on the linear space of polynomials in several variables with real coefficients. We study the so-called Uvarov and Christoffel modifications obtained by adding to the moment functional a finite set of mass points, or by multiplying it times a polynomial of total degree 2, respectively. Orthogonal polynomia...

the main purpose of this article is to present an approximate solution for the two-dimensional nonlinear volterra integral equations using legendre orthogonal polynomials. first, the two-dimensional shifted legendre orthogonal polynomials are defined and the properties of these polynomials are presented. the operational matrix of integration and the product operational matrix are introduced. th...

This paper deals with symmetric and non-symmetric polynomial perturbations of symmetric quasi-de nite bilinear functionals. We establish a relation between the Hessenberg matrices associated with the initial and the perturbed functionals using LU and QR factorizations. Moreover we give an explicit algebraic relation between the sequences of orthogonal polynomials associated with both functionals.

The system of Gegenbauer polynomials fC n (x); n = 0; 1; : : :g is a classical family of polynomials orthogonal with respect to the weight function ! (x) = (1?x 2) ? 1 2 on the support interval ?1; +1]. Integral functionals of Gegenbauer polynomials with kernel f(x))C n (x)] 2 ! (x), where f(x) is an arbitrary function which does not depend on n or , are considered in this paper. Firstly, a gen...