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

TL;DR

This paper derives tight, closed-form bounds and approximations for the Nuttall Q-function and incomplete Toronto function, facilitating their computation in wireless communication and engineering applications.

Contribution

It introduces novel analytic bounds and approximations for the Nuttall Q-function and incomplete Toronto function, which are not readily available in standard software.

Findings

Derived tight upper bounds for truncation errors.

Provided closed-form approximations that are accurate under certain conditions.

Enhanced computational handling of these special functions in applications.

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.