Hello

In the derivation of the cornish fisher expansion, the following equation is obtained: $$ \sum_{n=2}^{\infty} b_n H_{n-1}(x_\alpha) = \sum_{j=1}^{\infty}\frac{(x_\alpha - z_\alpha)^j}{j!}H_{j-1}(x_\alpha), $$ where $H_n(x)$ are the Hermite polynomials. To complete the derivation, this equation is used to express $x_\alpha$ in terms of $z_\alpha$. I was wondering how to go about doing this? (Kotz, Balakrishnan and Johnson state in Continuous Univariate Distributions, state that this can be achieved through tedious algebra using this equation.) I believe it might have something to do with series inversion. Quite stuck, any direction would be appreciated.

Thanks