Useful Results for Computing the Nuttall${-}Q$ and Incomplete Toronto Special Functions

Abstract: This work is devoted to the derivation of novel analytic results for special functions which are particularly useful in wireless communication theory. Capitalizing on recently reported series representations for the Nuttall $Q{-}$function and the incomplete Toronto function, we derive closed-form upper bounds for the corresponding truncation error of these series as well as closed-form upper bounds that under certain cases become accurate approximations. The derived expressions are tight and their algebraic representation is rather convenient to handle analytically and numerically. Given that the Nuttall${-}Q$ and incomplete Toronto functions are not built-in in popular mathematical software packages, the proposed results are particularly useful in computing these functions when employed in applications relating to natural sciences and engineering, such as wireless communication over fading channels.