Beta distribution central moment I am studying the note "On the sub-Gaussianity of the Beta and Dirichlet distributions" written by Olivier Marchal and Julyan Arbel(2017) which is available here.
In page 5, it says
$$ E\Bigg[\Big(X-\frac{1}{2}\Big)^{2j}\Bigg]=\frac{(2j)!}{2^{2j}j!}\frac{\Gamma(2\alpha)\Gamma(\alpha+j)}{\Gamma(\alpha)\Gamma(2(\alpha+j))}=\frac{(2j)!}{2^{2j}j!}\frac{(\alpha)_j}{(2\alpha)_{2j}}$$
where $X \sim Beta(\alpha, \alpha)$, and $(\alpha)_j=\alpha(\alpha+1)...(\alpha+j-1)=\frac{\Gamma(\alpha+j)}{\Gamma(\alpha)}$.
Can someone tell me what algebra can make the expectation as RHS.
 A: The Beta$(\alpha,\alpha)$ density, defined on the interval $[0,1],$ is
$$f_\alpha(t) = \frac{1}{B(\alpha,\alpha)}\, t^{\alpha-1}(1-t)^{\alpha-1}$$
where the normalizing constant is the Beta function, known to equal
$$B(\alpha,\alpha) = \int_0^1 t^{\alpha-1}(1-t)^{\alpha-1}\,\mathrm{d}t = \frac{\Gamma(\alpha)\,\Gamma(\alpha)}{\Gamma(2\alpha)}.\tag{*}$$
The substitution $t = (1+\sin\theta)/2$ is a one-to-one mapping from $[-\pi/2,\pi/2]$ to $[0,1]$ that re-expresses the expectation as
$$\begin{aligned}
E_{f_\alpha}\left[\left(X-\frac{1}{2}\right)^{2j}\right] &= \int_0^1 \left(t-\frac{1}{2}\right)^{2j}\,f_\alpha(t)\,\mathrm{d}t\\
&=\frac{1}{2^{2j+2\alpha-1}B(\alpha,\alpha)}\int_{-\pi/2}^{\pi/2} \cos^{2\alpha-1}(\theta)\,\sin^{2j}(\theta)\,\mathrm{d}\theta\\
&= \frac{B\left(j+\frac{1}{2},\alpha\right)}{2^{2j+2\alpha-1}B(\alpha,\alpha)}.
\end{aligned}$$
(The trigonometric integral is well known; see, for instance, Whittaker & Watson 12.42.)
Use the formula $(*)$ and the Gamma recurrence
$$\Gamma(z+1) = z\Gamma(z)$$
to re-express this in the form given in the question.

Reference
E. T. Whittaker  & G. N. Watson, A Course of Modern Analysis. Fourth Edition (1927).
A: A bit more precise answer: the central moments of beta distribution can be expressed in terms of the hypergeometric function.
https://mathworld.wolfram.com/BetaDistribution.html
The setup $\alpha=\beta$ from the question follows then as in such a case formulas greatly simplify (well-known special cases).
