Closed formula for $[\hat{H},[\hat{H},...[\hat{H},\hat{c}^\dagger_{\mu,\sigma}]]...]]$ Given the interaction part of a general many-body hamiltonian, 
$$\hat{H}=\sum_{\alpha, \beta,\gamma,\delta,\sigma,\sigma^\prime}O_{\alpha,\gamma,\sigma}^{\beta,\delta,\sigma^\prime}\hat{c}_{\alpha,\sigma}^\dagger \hat{c}^\dagger_{\beta,\sigma^\prime} \hat{c}_{\delta,\sigma^\prime}\hat{c}_{\gamma,\sigma},$$
I define 
$$\hat{C}_1=[\hat{H},\hat{c}_{\mu,\sigma}],$$
$$\hat{C}_n=[\hat{H},\hat{C}_{n-1}] \ \text{for every} \ n \gt 1.$$
I was wondering if there's any closed formula for $C_n,$ or at least a nicer recursion relation. $C_1$ and $C_2$ are easy to compute. I've also computed $C_3,$ which is already quite a lenghty and messy calculation!  But I haven't seen any pattern, and I haven't been able to write down a general expression for $C_n.$ Is this something known? Is it possible to do this?
 A: $\textbf{Hint:}$ This is a twisted way to put perturbation theory in Heisenberg picture.


*

*Define a set of generating operators as 
$$\hat{G}_{\mu \sigma}\left(z\right)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\underbrace{[\hat{H},[\hat{H},...[\hat{H},\hat{c}^\dagger_{\mu,\sigma}]]...]]}_{n^{\text{th}}-\text{order nested commutator}}~\stackrel{\textbf{BCH}}{=}~e^{z\hat{H}}\hat{c}^{\dagger}_{\mu,\sigma}e^{-z\hat{H}}$$
and 
$$\hat{\bar{G}}_{\mu \sigma}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}\underbrace{[\hat{H},[\hat{H},...[\hat{H},\hat{c}{}_{\mu,\sigma}]]...]]}_{n^{\text{th}}-\text{order nested commutator}}~\stackrel{\textbf{BCH}}{=}~e^{z\hat{H}}\hat{c}_{\mu,\sigma}e^{-z\hat{H}}.$$

*Notice
$$\frac{\mathrm{d}}{\mathrm{d}z}\hat{G}_{\mu \sigma}(z)~=~[\hat{H},\hat{G}_{\mu \sigma}(z)]$$
and 
$$\frac{\mathrm{d}}{\mathrm{d}z}\hat{\bar{G}}_{\mu \sigma}(z)=[\hat{H},\hat{\bar{G}}_{\mu \sigma}(z)].$$

*Use commutation/anti-commutation relations.

*Integrate the resulting equations between $\left[0,z\right]$ and iterate to find Taylor expansion coefficients of $\hat{G}_{\mu \sigma}(z)$ and $\hat{\bar{G}}_{\mu \sigma}(z)$.

*Check if you can find any pattern.
