Motivation
I have a prior on a random variable $X\sim \beta^\prime(\alpha,\beta)$ that I would like to use in JAGS. JAGS does not support the $\beta^\prime$ distribution, but it does support the $F$ distributions, and the F distribution is related to the $\beta^\prime(\alpha,\beta)$ like this:
$$\text{if } X\sim \beta^\prime(\alpha,\beta) \text{ then } X\frac{\alpha}{\beta}\sim F(2\alpha, 2\beta)$$
Question
Is there a transformation $$c,d = f(\alpha, \beta)$$ such that:
$$\text{if } X\sim \beta^\prime(\alpha,\beta)\text{ then } X\sim F(c, d)?$$
What is an appropriate analytical approach or solution to this problem?
Current Approach
My solution is to use simulation. Although sufficient for my application, a formal solution would be more satisfying.
set.seed(0)
alpha <- 2
beta <- 4
Y <- rf(100000, 2*alpha, 2*beta) * ( beta / alpha )
parms <-signif(fitdistr(Y, 'f', start = list(df1=1, df2=2))$estimate,2)
Update whuber's answer states that there is no general transformation.