Can someone explain why the following identity regarding the modified Bessel function of the first kind holds?
$$\int_{\mathbb{S}^{p-1}} e^{\kappa \mu^T\mathbf{x}}d\mathbf{x} = B\left(\frac{p-1}{2}, \frac{1}{2}\right)^{-1} \int_{-1}^{1} e^{\kappa t} (1 - t^2)^{(p-3)/2}dt = \Gamma\left(\frac{p}{2}\right)\left(\frac{\kappa}{2}\right)^{1-p/2} I_{p/2-1}(\kappa)$$
where $\kappa > 0$, $\mathbb{S}^{p-1} = \{\mathbf{x} \in \mathbb{R}^p: \mathbf{x}^T\mathbf{x} = 1\}$ is the unit sphere, $\mu$ is a unit vector such that $\mu^T\mu = 1$ (i.e. $\mu \in \mathbb{S}^{p-1}$), $B(\cdot,\cdot)$ is the beta function and $I_{\nu}(x)$ is the modified Bessel function of the first kind.
While the second equality is simple to derive from the definition of $I_{\nu}(x)$ (see for example https://dlmf.nist.gov/10.32), I do not understand how to perform the change of variable in the first equality.
For reference, the above chain of equalities is reported in equation (9.3.6) in Mardia and Jupp (2000), Directional Statistics.
Rotate the sphere or choose coordinate axes so that $\boldsymbol\mu$ becomes the direction of a coordinate axis (say $t$) and now you have the orthogonal $(p-2)$-sphere of radius $\sqrt{1-t^2}$ where the integrand takes the same value $e^{\kappa t}$: $$ \int_{\mathbb{S}^{p-1}}e^{\kappa {\boldsymbol \mu}^T\mathbf{x}}\,\mathrm{d}\mathcal{H}^{p-1}(\mathbf{x}) = \int_{-1}^1 e^{\kappa t}\left(\int_{\mathbb{S}^{p-2}(\sqrt{1-t^2}) }\,\mathrm{d}\mathcal{H}^{p-2}\right)\,\frac{\mathrm{d}t}{\sqrt{1-t^2}} $$ and we know $\mathcal{H}^{p-2}(\mathbb{S}^{p-2})=\frac{2\pi^{(p-1)/2}}{\Gamma(\frac12(p-1))}$.
Edit: The difference between the factor $\mathcal{H}^{p-2}(\mathbb{S}^{p-2})$ and your $\mathrm{B}(\frac{p-1}{2},\frac12)^{-1}$ appears to come from whether you normalise the measure on $\mathbb{S}^{p-1}$ to $1$.