  This paper presents novel analytic expressions for the Rice $Ie{-}$function,
$Ie(k,x)$, and the incomplete Lipschitz-Hankel Integrals (ILHIs) of the
modified Bessel function of the first kind, $Ie_{m,n}(a,z)$. Firstly, an exact
infinite series and an accurate polynomial approximation are derived for the
$Ie(k ,x)$ function which are valid for all values of $k$. Secondly, an exact
closed-form expression is derived for the $Ie_{m,n}(a,z)$ integrals for the
case that $n$ is an odd multiple of $1/2$ and subsequently an infinite series
and a tight polynomial approximation which are valid for all values of $m$ and
$n$. Analytic upper bounds are also derived for the corresponding truncation
errors of the derived series'. Importantly, these bounds are expressed in
closed-form and are particularly tight while they straightforwardly indicate
that a remarkable accuracy is obtained by truncating each series after a small
number of terms. Furthermore, the offered expressions have a convenient
algebraic representation which renders them easy to handle both analytically
and numerically. As a result, they can be considered as useful mathematical
tools that can be efficiently utilized in applications related to the
analytical performance evaluation of classical and modern digital communication
systems over fading environments, among others.
