Maybe. I'm really not sure what all the variable and function names are supposed to represent.
As far as I could see, the moment-generating function
μ(x) has the Taylor series
μ(x)=k=0∑∞k!xkμk(x)(∗)with the
k-th moments
μk(x)=E(Xk).
Let's see where we get at if we take the moment-generating function from Wikipedia:
μ(x)=(b−xb)p(∗∗).The parameters
b,p are defined by the density function
f(x)=⎩⎪⎨⎪⎧Γ(p)bpxp−1e−bx0 if x>0 if x≤0of the Gamma-distribution
G(p,b) that are necessary to get the correct scaling factors since the integral over the density function has to equal
1.
My suspicion is that
β=b,α=p,r=k to match the various parameter names. Perhaps
β=p,α=b,r=k. However, I cannot be sure at this point of the consideration. The next step would be to develop the Taylor series of
μ(x) as defined in
(∗∗) and compare the coefficients with the terms in
(∗). I admit, I'm a bit lazy to perform all the differentiations, so I asked WA (
https://www.wolframalpha.com/input?i=Taylor+expansion+of+(b/(b-x))^p) to do it for me and got
μ(x)=(b−xb)p=k=0∑∞k!pk[log(b−xb)]kComparing the coefficients yields
μk(x)=(xp)k[log(b−xb)]k=[xplog(b−xb)]k.This leaves us with the task to compare
μk′(x)=dxd[xplog(b−xb)]k with βkΓ(α)Γ(α+k),assuming
r=k.
That's not easy to see and still possibly wrong since Wikipedia isn't the most reliable source plus I may have made mistakes. However, I will change the editor to do some calculations off-line.