Confidence-interval of continuous-Poisson distribution

I have searched the state of the art of continuous-Poisson distribution [1](normally known as) \begin{align} f(k,\lambda) = \frac{\lambda^k e^{-\lambda}}{\Gamma(k+1)},\lambda\in R^+, k\in (-1,\infty) \end{align} ($f(;)$ as the PDF, $\lambda$ as the distribution parameter and $k$ as the observations value) as well as confidence interval of the Poisson-distribution-parameter \begin{align} \frac{1}{2}\chi^2(\frac{\alpha}{2};2k)<\mu<\frac{1}{2}\chi^2(1-\frac{\alpha}{2};2k+2) \end{align} ($\mu$ as the mean, $\alpha$ as the confidence level, and $\chi^2(x;n)$ as the PDF of the Chi squared distribution with $n$ degree of freedom) for checking the confidence interval of the parameter of ($\lambda$) continuous-Poisson distribution. However, unfortunately I could only find this old paper [1] that provides only an implicit relation \begin{align} \int_k^\infty f(k,\hat{\mu}_1)dk=\frac{\alpha}{2},\\ \int_{l(\hat{\mu}_2)}^k f(k,\hat{\mu}_2)dk=\frac{\alpha}{2}. \end{align} ($\hat{\mu}_1$ and $\hat{\mu}_2$ as the lower and upper bounds of the confidence interval and $l(\lambda)$ as a function of the Poisson parameter that for all the $\lambda$ parameters, outputs so that $\int_{l(\lambda)}^\infty f(k,\lambda)dk = 1$ where $l(\lambda)>-1$). As it looks, regarding that their relation is implicit, they provided a table for confidence interval bounds of some $\mu$ values.

However, I am seriously searching for an explicit relation for the confidence bounds. Does anyone know or have any guide to an explicit relation for confidence interval of the parameter of the continuous-Poisson distribution?