# Moment generating function of Beta distribution [closed]

I need help with the proof of how this expression ${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}$ or this expression $M_X(t)=\sum_{k=0}^\infty \frac{\Gamma(p+k)\Gamma(p+q)t^k}{\Gamma(p+q+k)\Gamma(p)k!}$ is the moment generating function for the Beta distribution.

• I need help with the integral $\int _{-\infty}^{\infty} e^{tx} Beta(x,a,b)dx$ – José C May 13 '17 at 23:14
• For posterity, here is a link from ProofWiki providing a derivation of the MGF of a Beta random variable: proofwiki.org/wiki/… – Luiz Max Carvalho Feb 5 '20 at 20:32

The moment generating function of a Beta random variable $X$ is defined for any $t$ and it is
$$M_X(t) = \sum_{k=0}^{\infty} \frac{t^k}{k!}\frac{B(\alpha+k,\beta)}{B(\alpha,\beta)} = 1 + \sum_{k=1}^{\infty}\frac{t^k}{k!}\prod_{n=1}^{k-1}\frac{\alpha + n}{\alpha+\beta+n}$$
$$_1 F_1(\alpha,\alpha+\beta,t)=1 + \sum_{k=1}^{\infty}\frac{t^k} {k!}\prod_{n=1}^{k-1}\frac{\alpha + n}{\alpha+\beta+n}$$