Orthogonal polynomials with respect to modified Jacobi weight and corresponding quadrature rules of Gaussian type

Abstract

In this paper we consider polynomials orthogonal with respect to the linear functional L : P→ℂ, defined on the space of all algebraic polynomials P by Mathematical Equation Representative, where α,β > -1/2 are real numbers such that ℓ = |β - α| is a positive integer, and ζ ε ℝ\{0}. We prove the existence of such orthogonal polynomials for some pairs of α and ζ and for all nonnegative integers ℓ. For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations. For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered. Also, some numerical examples are included. © 2011 Global-Science Press.