We derive the infinite sum of the Lerch function in terms of the incomplete gamma function and the Lerch function. Special cases are evaluated in terms of fundamentals constants. All the results in this work are new.

In the light of recent experimental results of Jung et al. [Phys. Rev. A 75, 052707 (2007)], we have carried out relativistic distorted wave calculations for electron impact excitation of the ten higher-lying fine-structure levels of the $3p^5$ $5p$ configuration of argon from the lowest metastable states (the $J=0,2$ levels of the $3p^5$ $4s$ configuration). We compare our theoretical results with their experimental results and discuss the differences from the similar excitation to the $3p^5$ $4s$ levels from the same metastable states.