Recursively obtaining all-pole coefficients from complex cepstral coefficients?
If $h[n]$ is the impulse response of an all-pole minimum-phase filter of order $p$, then we have

$$\mathcal{Z}\{h[n]\}=H(z)=\frac{G}{1-\displaystyle\sum_{k=1}^pa_kz^{-k}}\tag{1}$$

where $G$ is some gain value, and $a_k$ are the predictor coefficients. Let $\hat{h}[n]$ be the complex cepstral coefficients of $h[n]$. If you know how to compute $\hat{h}[n]$ from $a_k$, then you're probably familiar with the following recursion:

$$\hat{h}[n]=a_n+\frac{1}{n}\sum_{k=1}^{n-1}ka_{n-k}\hat{h}[k],\quad 0&lt;n\le p\tag{2}$$

From $(2)$ you can formulate a recursion for $a_k$:

$$a_n=\hat{h}[n]-\frac{1}{n}\sum_{k=1}^{n-1}ka_{n-k}\hat{h}[k],\quad 1&lt;n\le p\tag{3}$$

with the initial value $a_1=\hat{h}[1]$.