We consider \(G(x)=F^{-1}\int^{x}_{0}(\Gamma(t))^{-1}dt,\) which can be associated with Ramanujan, and study also a generalization of this reciprocal gamma distribution. Thereby we used the functional relation \(f(x+1)=r(x)f(x)\) with r(x) e.g. rational.
Primarily, we wanted to find numerical values (to 30D) of population characteristics, but we found the moment generating function \(\phi(t)=\int^{\infty}_{0}\exp(-tx)/\Gamma(x)dx\) more and more interesting. Thus, we derived recurrence formulae for the moments using e.g. Bell and Stirling numbers.
We could use a formula by Ramanujan to obtain analytical expressions of \(\phi\) (t) and also we established connections with beta and the confluent hypergeometric function. We found some inequalities giving upper and lower bounds for \(\phi\) (t). The numerical calculation of \(\phi\) (t) in the interval \(0\leq t\leq 5\) to 6D was made using a lot of methods, but we could only rely on one quadrature formula in the whole interval to this accuracy together with Euler-Maclaurin expansion with step-length \(h={1\over2}\) and even less as given in a note by Wrigge (reviewed below).
The present tabulated values of the moments to 30D and abscissae and weights for the 14 and 15 points quadrature formulae used to 25S among other results.