After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one.
Let $V$ be a $Beta(\alpha,1)$ random variable, with density $$ g(v) = \alpha v^{\alpha-1}, $$ with $v\in[0,1]$. Then the Moment Generating Function is a confluent hypergeometric function, with a number of possible representations: $$ MGF_V(\alpha,1,t) = \int_0^1 e^{t v}g(v)dv, $$
$$ MGF_V(\alpha,1,t) = 1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+r+1} \right) \frac{t^k}{k!} $$
$$ MGF_V(\alpha,1,t) = \alpha (-t)^{-\alpha} (\Gamma(\alpha) - \Gamma(\alpha,-t)) $$ Where $\Gamma(x,t)$ is the incomplete Gamma function. There are many more, apparently. My interest is in the Legendre Transform of $\log MGF_V$: $$ I(s) = \sup_t t s - \log MGF_V(t) $$ This is a cost function for an optimal control problem. I need to evaluate it. The first order condition of this optimization is $$ s- \frac{MGF'_V(t)}{MGF_V(t)} =0 $$ There doesn't seem to be a convenient way to solve this. I've done so numerically in Matlab using their implementations of the various different functions that can be used to calculate $MGF_V$, but have lots of numerical instability as the parameter $\alpha$ and $s$ vary. Basically, my question is, is there a convenient way to express or calculate $$ \frac{MGF'_V(t)}{MGF_V(t)} $$ that I have overlooked? Some of the forums above are such that the derivative is easy to calculate - for example, using the representation as $\Gamma$ functions, $$ MGF'_V(t) = \frac{\alpha e^t}{t} +\alpha^2(-t)^{-\alpha-1}(\Gamma(\alpha) - \Gamma(\alpha,-t), $$ so that the ratio of interest is $$ \frac{MGF'_V(t)}{MGF_V(t)} = \frac{-\alpha + \frac{e^t (-t)^\alpha}{\Gamma(\alpha) - \Gamma(\alpha,-t)}}{t} $$ which is no harder to calculate than $MGF_V(t)$, but it is not obvious to me how to solve $$ s = \frac{-\alpha + \frac{e^t (-t)^\alpha}{\Gamma(\alpha) - \Gamma(\alpha,-t)}}{t} $$ for $t$ other than numerically, and I often have trouble with the implementations of $\Gamma$ in Matlab.