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It's well-known (see e.g. the Wikipedia article) that the Lambert $W$ function can be used to solve equations of the form $$x = a e^{b x}$$ by setting $$x = - \frac{W(-ab)}{b}.$$ This follows from the fact that $W$ satisfies $W(x) = x e^{-W(x)}$ (in fact, $W$ solves a more general family of equations, but I'm going to keep things simple for the sake of the question).

In my research in algebraic combinatorics I've come across a nice two-variable generating function $F(x_1, x_2)$ that, among other things, satisfies $F = x_1 \exp(- x_2 \exp(F))$. Then $F$ can be used to solve equations of the form $$ x = a \exp\big(b \exp(c x)\big) \label{1}\tag{$\star$}$$ by setting $$ x = \frac{F(ac, -b)}{c}.$$ (Again, the story is more general, but this is the key fact for my question.)

My question is: are equations similar to \eqref{1} of any interest? If yes, do they have a name, and what problems do they solve?

[also tagging complex analysis and differential equations since I know $W$ comes up there, and $F$ is in many ways a generalization of $W$]

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    $\begingroup$ Isn't that just a matter of using Lagrange reversion theorem to obtain $F$? $\endgroup$ Commented Feb 4 at 12:31
  • $\begingroup$ Could be (though I don't see it immediately), but my question is more about whether an equation of the form indicated has ever come up in the literature with regards to a specific problem. $\endgroup$ Commented Feb 4 at 18:59

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Yes, equations similar to $(\star)$ are of interest. They arise in counting bicolored trees. The coefficients of $A(x,y)$ satisfying $A(x,y) = \exp(x\exp( yA(x,y)))$ appear in the OEIS as sequence A072590. With a different normalization they appear as sequence A161552.

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  • $\begingroup$ Thanks! Do you know if that's the only place where it comes up? I'm asking since $y = Ae^{B y}$ comes up in some physics models. $\endgroup$ Commented Feb 7 at 22:03
  • $\begingroup$ No, I don't know of any other place other than the enumeration of certain kinds of trees. There is a book by István Mező, about generalizations of the Lambert W function: The Lambert W function—its generalizations and applications, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2022. I have not seen this book but there might be something relevant to your question in it. $\endgroup$ Commented Feb 7 at 23:34
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$$x=ae^{be^{cx}}$$

Equations of this kind can be solved by hyper-Lambert of Galidakis et al.:

$$xe^{-be^{cx}}=a$$ $$\frac{1}{c}G(-b;cx)=a$$ $$G(-b;cx)=ac$$ $$x=\frac{1}{c}HW(-b;ac)$$
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For $F(x_1,x_2)$:

$$y=x_1e^{-x_2e^y}$$ $$ye^{x_2e^y}=x_1$$ $$G(x_2;y)=x_1$$ $$y=HW(x_2;x_1)$$ $$F(x_1,x_2)=HW(x_2;x_1)$$

[Galidakis 2007] gives a numerical algorithm and a Maple procedure for calculating the values of $HW$.

[Galidakis 2007] Galidakis, I. N.: On some applications of the generalized hyper-Lambert functions. Complex Variables and Elliptic Equations 52 (2007) (12) 1101-1119

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