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Solving a particular delay PDE $\partial_q f(q,s-1) = -\sqrt{s(2+s)}f(q,s)$

Inspired by your comparison with the Hurwitz zeta function, which has the integral representation $$ \zeta (s,a)={\frac {1}{\Gamma (s)}}\int _{0}^{\infty }{\frac {x^{s-1}e^{-ax}}{1-e^{-x}}}\mathrm{d}x …
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