Let $d$ be an positive integer and $p\geq 0$. Define a quantity
$$\mu_{d,p} = \frac{\Gamma\big(\frac{d}{2}\big)\Gamma\big(\frac{p+1}{2}\big)}{\sqrt{\pi}\Gamma\big(\frac{d+p}{2}\big)}.$$
How to estimate the asymptotic bounds of $\mu_{d,p}$ as $d\to\infty$?
Let $v$ be a random variable which is uniformly at $S^{d-1}$, the unit sphere in $\mathbb{R}^d$, and $u$ is a vector in $S^{d-1}$. How to calculate the following expectation?
$$ \mathbb{E}_v|\langle v,u\rangle|^p. $$
These two quantities appears in a paper I am reading, which says that by a standard calculation
$$ \mathbb{E}_v|\langle v,u\rangle|^p = \frac{2}{B\big(\frac{1}{2},\frac{d-1}{2}\big)}\int_0^1t^p(1-t^2)^{(d-3)/2}dt =\mu_{d,p}. $$ I need more details about how those two equations hold.
My goal is to estimate the asymptotic bounds of $\mathbb{E}_v|\langle v,u\rangle|^p$ so these are the two technical questions I need to solve.
It is possible that I misunderstood the question.
Concerning $$\int_0^1t^p\,(1-t^2)^{\frac{d-3}2}\,dt=\frac{\Gamma \left(\frac{d-1}{2}\right) \Gamma \left(\frac{p+1}{2}\right)}{2 \Gamma \left(\frac{d+p}{2}\right)}$$ and the relation between beta and gamma functions effectively leads to $$\mathbb{E}_v|\langle v,u\rangle|^p = \frac{2}{B\big(\frac{1}{2},\frac{d-1}{2}\big)}\int_0^1t^p\,(1-t^2)^{\frac{d-3}2}\,dt=\frac{\Gamma \left(\frac{d}{2}\right) \Gamma \left(\frac{p+1}{2}\right)}{\sqrt{\pi } \,\,\Gamma \left(\frac{d+p}{2}\right)}=\mu_{d,p}$$ For large values of $p$, taking logarithms, using Stirling approximation and continuing with Taylor series using $\mu_{d,p}=e^{\log(\mu_{d,p})}$, we end with $$\mu_{d,p}\sim\frac{\Gamma \left(\frac{d}{2}\right) }{\sqrt{\pi }}\left(\frac{p}{2}\right)^{\frac{d-1}{2}}\exp\left(-\frac{(d-1)^2}{4 p} +\frac{d(d-1) (d-2) }{12 p^2}+\cdots\right)\sim\frac{\Gamma \left(\frac{d}{2}\right) }{\sqrt{\pi }}\left(\frac{p}{2}\right)^{\frac{d-1}{2}}$$