Define
$p_\delta^{\pm,q}\left(z\right)=\prod\limits_{n=1}^\infty\left(1\pm e^{-2z\sqrt{\frac{\delta^2}4+\pi^2\left(n-\frac q2\right)^2}}\right)$
and
$P_\delta^{\pm,q}\left(z\right)=\prod\limits_{n\in\mathbb Z,\,n\equiv q\,\left(\text{mod }2\right)}\left(1\pm e^{-z\sqrt{\delta^2+\pi^2n^2}}\right)$
where $z\in\mathbb C$ with $\mathrm{Re}\left(z\right)>0$ and $\delta\in\mathbb R$ and $q=0,1$. Can any of these functions be expressed in terms of some known special functions or in terms of an integral? If possible, both representations would be great. Also, what properties do these functions have? Are they analytic in $z$ and are the smooth at $\delta=0$? Any progress made on any of the functions would be really helpful. Please do comment/answer even if there's a partial answer to any of these questions.