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Numerical construction of the generalized Hermite polynomials

Milovanović Gradimir V. (Faculty of Electronic Engineering - Department of Mathematics, SCG)
Cvetković Aleksandar S. (Faculty of Electronic Engineringh - Department of Mathematics, SCG)

In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x − z|γ exp(−x2) on R, where zєR and γ > − 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.

Keywords: Orthogonal polynomials, monic polynomials, norm, inner product, generalized Hermite weight, three term recurrence relation, numerical quadrature, nonlinear recurrence relations, asymptotic expansion