An orthogonality relation for the Whittaker functions of the second kind of imaginary order

TL;DR

The paper provides a direct, distribution-theoretic derivation of the orthogonality of Whittaker functions of imaginary order $W_{\kappa,i\mu}(x)$ on $(0,\infty)$ with weight $x^{-2}$. The main result is the explicit distributional formula $\int_0^\infty dx\; W_{\kappa,i\mu}(x) W_{\kappa,i\mu'}(x)/x^2 = \frac{\pi^2}{\mu\sinh(2\pi\mu) \Gamma(1/2-\kappa+i\mu) \Gamma(1/2-\kappa-i\mu)} [\delta(\mu-\mu')+\delta(\mu+\mu')]$, valid for real $\mu,\mu'$. For $\kappa=0$ this reduces to the orthogonality of Macdonald functions $K_{i\mu}$, revealing connections to the Kontorovich–Lebedev transform; the paper also discusses specializations when $\kappa$ is integer or half-integer and alternative delta-function formulations. This tightens the understanding of spectral orthogonality for imaginary-order Whittaker functions and provides a tool for applications in related transform theories.

Abstract

An orthogonality relation for the Whittaker functions of the second kind of imaginary order, $W_{κ,\mathrm{i}μ}(x)$, with $μ\in\mathbb{R}$, is investigated. The integral $\int_{0}^{\infty}\mathrm{d}x\: x^{-2}W_{κ,\mathrm{i}μ}(x)W_{κ,\mathrm{i}μ'}(x)$ is shown to be proportional to the sum $δ(μ-μ')+δ(μ+μ')$, where $δ(μ\pmμ')$ is the Dirac delta distribution. The proportionality factor is found to be $π^{2}/[μ\sinh(2πμ)Γ({1/2}-κ+\mathrm{i}μ) Γ({1/2}-κ-\mathrm{i}μ)]$. For $κ=0$ the derived formula reduces to the orthogonality relation for the Macdonald functions of imaginary order, discussed recently in the literature.