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. (read more)