Let $M$ be a Riemannian manifold, fix some $y\in M$ and let $U_y\subset M$ be a normal neighborhood of $y$. We are given a vector bundle $E$ with base space $M$ together with a covariant derivative $\nabla$. Lastly, suppose that we are given a sequence $\Phi_n\in\Gamma(U_y,E)\otimes E_y^*$ - i.e. $\forall n,x:\Phi_n(x)\in L(E_y,E_x)$ - satisfying the following relations: $$\nabla_{r\partial_r}\Phi_0=0$$ $$(\nabla_{r\partial_r}+n)\Phi_n=-B\Phi_{n-1}$$ Here $B:\Gamma(M,E)\to\Gamma(M,E)$ is some operator and $\partial_r$ is the radial vector field. In the following we slightly abuse notation and write $x=\exp_yx$ (i.e. the normal coordinates are left implicit).
Theorem $2.26$: Let $1\leq n$, then $$\forall x\in U_y:\Phi_0(x)^{-1}\Phi_n(x)=-\int_0^1t^{n-1}\Phi_0(tx)^{-1}(B\Phi_{n-1})(tx)dt$$ Unfortunately I don't understand the proof:
The hint is to consider the function$$\phi(t)=t^n\Phi_n(tx)$$ but I think that they actually meant $\phi(t):=t^n\Phi_0(tx)^{-1}\Phi_n(tx)$: As far as I understand, we want to use the fact that$$\Phi_0(x)^{-1}\Phi_n(x)=\phi(1)-\phi(0)=\int_0^1\frac{\mathrm d\phi}{\mathrm dt}(t)dt$$ and indeed the author claims that $$\nabla_{d/dt}\phi=-t^{n-1}(B\Phi_{n-1})$$ which kind of looks like the desired result, but I don't even know how the LHS of the last equation is defined. Is it just some fancy notation for the time derivative? If yes, how to prove the last equation? There are too many steps missing, I hope someone can elaborate.