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

Abstract

Explicit formulae are given for the Hilbert transform $$f_\mathbb{R} $$ w(t)dt/(t − x), where w is either the generalized Laguerre weight function w(t) = 0 if t ≤ 0, w(t) = t α e −t if 0 −1, x > 0, or the Hermite weight function w(t) = e −t 2, −∞ <#60; t <#60; ∞, and −∞ <#60; x <#60; ∞. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddle-point asymptotics, and series expansions. We also study the numerical stability of the three-term recurrence relation satisfied by the integrals $$f_\mathbb{R} $$ π n (t;w)w(t)dt/(t − x), n = 0 ,1 ,2 ,..., where π n (⋅w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n.