Bob, Many Thanks for the ideas and to Sakrejda as well.
If the required precision of Stan is only 1e-8 then I would suggest to think of using
a good approximation of the log-gamma function. And this would lead me to another
question wouldn’t it be beneficial to have approximations in Stan?
Why did Ben use exp(-745) as termination criteria in this Conway-Maxwell-Poisson
Code? http://discourse.mc-stan.org/t/conway-maxwell-poisson-distribution/2370
Approx. of Gamma Functions
http://www.rskey.org/CMS/index.php/the-library/11
https://arxiv.org/abs/1003.6020
Approx. of Mittag-Lefflers function
This is not an easy task
http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=95FB100080A764194C9BE37F0FDD4FCD?doi=10.1.1.706.6903&rep=rep1&type=pdf
https://www.icp.uni-stuttgart.de/~hilfer/publikationen/pdfa/ITSF-2006-17-637-a.pdf