In Partial Differential Equations by Evans there's this one identity that is used to solve the Euler-Poisson-Darboux equation for odd $n \geq 3$:
$$ \left( \frac{1}{r} \frac{d}{dr} \right)^{k-1} \left( r^{2k-1} \phi(r)\right)=\sum_{j=0}^{k-1} \beta^k_j r^{j+1} \frac{d^j \phi(r)}{dr^j}$$
where $\phi: \mathbb{R} \rightarrow \mathbb{R}$ is $C^{k+1}$, the $\beta_j^k$ are independent of $\phi$ and $$\beta_0^k=1 \cdot3 \cdot 5 \cdots (2k-1) \,.$$
The proof (by induction) is omitted and my classmates and I are stumped. Has anyone else done this proof before, or seen it done? I've looked everywhere but the proof is always omitted in lecture notes and textbooks.