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Dumont's paper "Pics de cycle et derivees partielles" gives, on p. 38, the continued fraction for an o.g.f. of the bivariate Eulerian polynomials as

$$x + \sum_{n \geq 1} A_n(x,y)\; u^n $$

so the question becomes what is the continued fraction for an e.g.f. given the continued fraction of its Laplace transform, or shifted o.g.f.?

Dumont's paper "Pics de cycle et derivees partielles" gives, on p. 38, the continued fraction for an o.g.f. of bivariate symmetric Eulerian polynomials $\bar{E}_n(x,y)$ related to mine by $\bar{E}_n(x,y) = xy \cdot E_n(x,y)$. Dumont's o.g.f. is

$$x + \sum_{n \geq 1} \bar{E}_n(x,y)\; u^n $$

but an e.g.f. is used in the compositional inversions above. This begs the questions of how continued fractions of o.g.f.s and their associated e.g.f.s are related and also the continued fractions of the compositional inverses of the two generating functions.

$$x + \sum_{n \geq 1} A_n(x,y) $$

$$x + \sum_{n \geq 1} A_n(x,y)\; u^n $$

$$ \begin{split} & = \frac{x}{1-\dfrac{yu }{1-\dfrac{xu}{1-\dfrac{ 2yu}{\ddots}} } \; \;}\\ &= SFC[\; x, \; \frac{yu}{xu}, \; 2 \frac{yu}{xu}, \; 3 \frac{yu}{xu},..] \end{split} $$

$$ \begin{split} & = \frac{x}{1-\dfrac{yu }{1-\dfrac{xu}{1-\dfrac{ 2yu}{\ddots}} } \; \;}\\ &= SFC[\; x, \; \frac{yu}{xu}, \; \frac{2yu}{2xu}, \; \frac{3yu}{3xu},..] \end{split} $$