Moment generating function of Beta distribution

Question

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.

Answer 1

The moment generating function of a Beta random variable $X$ is defined for any $t$ and it is

$$M_X(t) = \sum_{k=0}^{\infty} \frac{t^k}{k!}\frac{B(\alpha+k,\beta)}{B(\alpha,\beta)} = 1 + \sum_{k=1}^{\infty}\frac{t^k}{k!}\prod_{n=1}^{k-1}\frac{\alpha + n}{\alpha+\beta+n}$$

The above formula for the moment generating function might seem impractical to compute, because it involves an infinite sum as well as products whose number of terms increase indefinitely. However, the function

$$_1 F_1(\alpha,\alpha+\beta,t)=1 + \sum_{k=1}^{\infty}\frac{t^k} {k!}\prod_{n=1}^{k-1}\frac{\alpha + n}{\alpha+\beta+n}$$

is a function, called Confluent hypergeometric function of the first kind, that has been extensively studied in many branches of mathematics. Its properties are well-known and efficient algorithms for its computation are available in most software packages for scientific computation.