Computing the Hilbert Transform of the
Generalized Laguerre and Hermite Weight Functions

by Walter Gautschi, Purdue University, West Lafayette IN, USA
Jörg Waldvogel, Swiss Federal Institute of Technology ETH,
CH-8092 Zürich, Switzerland


The Hilbert transform W(x) of the function w(t) is defined as the Cauchy principal-value integral of w(t)/(t-x) over
the real t-axis. We give explicit formulae for the W(x) in the cases of the generalized Laguerre weight function,
w(t) = t^a exp(-t) (1+sign(t))/2, and of the Hermite weight function, w(t) = exp(-t^2). Furthermore, several numerical evaluation schemes are discussed, based on various representations of the objects under consideration. In this connection we study the numerical stability of the three-term recurrence relation satisfied by certain related integrals involving the Laguerre or the Hermite polynomials.

The complete paper appeared in BIT 41, 2001, 490-503.
Download a preliminary version (15 pages, Fig. 3 missing, Fig. 1 improved): hilbtransf.pdf