Integral and series representations of the Nuttall function (Q6591526)

The purpose of this paper is to prove two new integral representation formulae for the Nuttall function \(Q_{\mu,\nu} (a, b)\). Connections with the important Marcum \(Q\) and the Toronto functions are pointed out. The main theorem gives a surprisingly simple relation between Marcum \(Q\)-function and the Nuttall function.\N\NThe connection between the Nuttall function and the complementary error function and Hermite polynomials by \textit{Yu. A. Brychkov} [Integral Transforms Spec. Funct. 25, No. 1, 34--43 (2014; Zbl 1303.33020)] is used to derive a simplified version of a previous formula. The Kampé de Fériet function is used to derive a formula for a suitable parametrization of the Toronto function. The proofs use Laplace transform of the generalized hypergeometric function, Kummer function, the general Leibniz rule. Finally, central processing unit times taken to compute integral representation for the Nuttall function are shown in diagrams.