A new special function related to a discrete Gauss-Poisson distribution and some physics of the cell model with Curie-Weiss interactions

Inspired by previous studies in statistical physics [see, in particular, Kozitsky at al., A phase transition in a Curie-Weiss system with binary interactions, Condens. Matter Phys. 23, 23502 (2020)] we introduce a discrete Gauss-Poisson probability distribution function \begin{equation}\label{GPD}\tag{A1} p_{GP}(n ;z,r)=\left[R(r;z)\right]^{-1}\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2} \end{equation} with support on $\mathbb N_0$ and parameters $z\in\mathbb R$ and $r\in\mathbb R_+$. The probability mass function $p_{GP}(n ;z,r)$ is normalized by the special function $R(r;z)$, given by the infinite sum \begin{equation}\label{R}\tag{A2} R(r;z)=\sum_{n=0}^\infty\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2}, \end{equation} possessing extremely intersting mathematical properties. We present an asymptotic estimate $R^{(\rm as)}(r;z\gg1)$ for the function $R(r;z)$ with large arguments $z$, along with similar formulas for its logarithm and logarithmic derivative. These functions exhibit very interesting oscillatory behavior around their asymptotics, for parameters $r$ above some threshold value $r^*$. Some implications of our findings are discussed in the context of the Curie-Weiss cell model of simple fluids.