d-fold Hermite-Gauss quadrature

We extend results presented in Gustafon and Hagler (J. Comput. Appl. Math. 105 (1999) 317-326); Hagler (Ph.D. Thesis, University of Colorado, 1997; J. Comput. Appl. Math. 104 (1999) 163-171; Hagler et al. (Lecture Notes in Pure and Applied Mathematics Series, Vol. 1999, Marcel Dekker, New York, 1998, pp. 187-208) by giving a construction of systems of orthogonal rational functions from systems of orthogonal polynomials and explicating the (2(d) n)-point d-fold Hermite-Gauss quadrature formula of parameters y, lambda > 0: integral (infinity)(-infinity) f(x)e(-[v[d](gamma,lambda)(x)]2) dx = Sigma (2dn)(k=1)f(h(d,n,k)((gamma,lambda)))H-d,n,k((gamma,lambda)) + E-d,n((gamma,lambda))[f(x)], where v([d](gamma,lambda))(x) is the d-fold composition of v((gamma,lambda))(x) = (1/lambda)(x - y/x) and where the abscissas h(d,n,k)((y,lambda)) and weights H-d,n,k((y,lambda)) are given recursively in terms of the abscissas and weights associated with the classical Hermite-Gauss quadrature. Error analysis, tables of numerical values for nodes, and examples and comparisons are included. (C) 2001 Published by Elsevier Science B.V.