  This article provides novel analytical results for the Rice function, the
incomplete Toronto function and the incomplete Lipschitz-Hankel Integrals.
Firstly, upper and lower bounds are derived for the Rice function, $Ie(k,x)$.
Secondly, explicit expressions are derived for the incomplete Toronto function,
$T_{B}(m,n,r)$, and the incomplete Lipschitz-Hankel Integrals of the modified
Bessel function of the first kind, $Ie_{\mu,n}(a,z)$, for the case that $n$ is
an odd multiple of 0.5 and $m \geq n$. By exploiting these expressions, tight
upper and lower bounds are subsequently proposed for both $T_{B}(m,n,r)$
function and $Ie_{\mu,n}(a,z)$ integrals. Importantly, all new representations
are expressed in closed-form whilst the proposed bounds are shown to be rather
tight. Based on these features, it is evident that the offered results can be
utilized effectively in analytical studies related to wireless communications.
Indicative applications include, among others, the performance evaluation of
digital communications over fading channels and the information-theoretic
analysis of multiple-input multiple-output systems.
