Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ is real, ($ n \ge 2$) ? And how to prove this?
I've a strange observation of periodicity which I would like to explain/derive/prove analytically.
Consider the triangle of eulerian numbers E (ideally of infinite size, row and col-indices beginning at zero, with elements $ \small e_{r,c}$)
$ \qquad E = \small \begin{array} {rrrrr} 1 & . & . & . & . & . \\ 1 & 0 & . & . & . & . \\ 1 & 1 & 0 & . & . & . \\ 1 & 4 & 1 & 0 & . & . \\ 1 & 11 & 11 & 1 & 0 & . \\ 1 & 26 & 66 & 26 & 1 & 0 \end{array} $
Assume some angular parameter $\varphi$, the associated complex number from the unit-circle $z=z_\varphi= \exp(i \varphi ) \qquad z \ne 1 $, $ \small {\varphi \over \pi} $ not necessarily rational.
Now consider the Eulerian polynomials, whose coefficients are taken from a row of E, $ \small E(z,r) = \sum_{c=0}^\infty ( e_{r,c} \cdot z^c ) $
Then compute $ \small \zeta_\varphi(-n) = {z \over (1-z)} \cdot E(z,n) \cdot (1-z)^{-n} $ .
Observation: I observe, that for $n>2$ the $ \small \zeta_\varphi(-n) $ lie either on the real or on the imaginary axis, in other words $ \small \zeta_\varphi(-n) ^2 $ are real..
Q: I seem to be missing just the key idea, so my question here is, how I could derive that behave analytically, given the description via the Eulerian triangle.
Addendum/Generalization: While it was already surprising, that this works with rational roots z of the complex unit, it seems to me even more interesting, that the observation holds for arbitrary $ \small \varphi \ne 0 $ and $ \small z_\varphi = \exp(I \cdot \varphi) \ne 1 $ .