Question Number 163548 by HongKing last updated on 07/Jan/22
$$\mathrm{Lambert}\:\mathrm{series}\:\mathrm{type}\:\mathrm{representation} \\ $$$$\mathrm{for}\:\:\sqrt{-\mathrm{1}}\:\:\mathrm{factorial} \\ $$$$\left(\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:\left(-\mathrm{i}^{\boldsymbol{\mathrm{i}}} \right)!\:=\:\mathrm{1}\:-\:\mathrm{2}\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(-\mathrm{1}\right)^{\boldsymbol{\mathrm{n}}} }{\mathrm{e}^{\boldsymbol{\pi}} \:\mathrm{n}^{\mathrm{2}} \:-\:\mathrm{1}} \\ $$