On the evaluation of an integral involving the Whittaker [formula omitted] function

Abstract

The topic of interest is the evaluation of the integral I(α;β)≔∫α+∞W1,β2(x)dxx2withα∈(0,∞)andβ∈ℂsuch thatW1,β(α)=0,where Wa,b(z) denotes the Whittaker W function. The question is brought about by a certain Sturm–Liouville problem: the (discrete) spectrum of the respective Sturm–Liouville operator is captured by β=βk, k∈N, and I(α;β) gives the (positive but finite) “length” of the kth non-normalized eigenfunction. The Sturm–Liouville problem is of importance in financial mathematics, mathematical physics, stochastic processes, and quickest change-point detection. We obtain a new analytic expression for I(α;β). The expression is in the form of a (convergent) series, which can be computed numerically via successive iteration of a homogeneous linear second-order difference equation. We also offer an exact closed-form solution to the difference equation. We conclude with a curious connection of our result to the latest developments in the theory of hypergeometric functions.