$\require{begingroup} \begingroup$ $\def\e{\mathrm{e}}\def\W{\operatorname{W}}\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$
\begin{align} \W_n\left((4n-1)\,\tfrac{\pi}2\right) +\W_{-n}\left((4n-1)\,\tfrac{\pi}2\right) &=0 \tag{1}\label{1} ,\\ \W_{n-1}(-(4n-3)\,\tfrac\pi2) +\W_{-n}(-(4n-3)\,\tfrac\pi2) &=0 \tag{2}\label{2} ,\\ \int_0^{(4n-1)\tfrac{\pi}2} -\tfrac12\,(\W_n(x)+\W_{-n}(x))\, dx &=(4n-1)\tfrac{\pi}2 \tag{3}\label{3} ,\\ \int_0^{(4n-3)\tfrac{\pi}2} -\tfrac12\,(\W_{n-1}(-x)+\W_{-n}(-x))\, dx &=(4n-3)\tfrac{\pi}2 \tag{4}\label{4} ,\\ \int_0^1 -\tfrac12\left(\W_n(\tfrac\pi2\,(4n-1)\,t) +\W_{-n}(\tfrac\pi2\,(4n-1)\,t)\right)\, dt &=1 \tag{5}\label{5} ,\\ \int_0^1 -\tfrac12\left(\W_{n-1}(-\tfrac\pi2\,(4n-3)\,t) +\W_{-n}(-\tfrac\pi2\,(4n-3)\,t)\right)\, dt &=1 \tag{6}\label{6} \end{align}
for $n>1$, where $\W_{n}$ is a complex-valued branch of the Lambert $\W$ function.
Question: is there known references where these expressions are mentioned?
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