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In 1758, Lambert solved the trinomial equation $x = q+x^m$ by giving a series development for $x$ in powers of $q$ as stated in On the Lambert W function (2nd page).

Looking for the series solution of $x = q+x^m$ and the detailed process by which it was solved.

(I have done a lot of searching but couldn't find it anywhere.)

Amit Hasan
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    The paper you link has a citation of Lambert's 1758 paper. Have you checked that? – Chappers Jul 14 '17 at 17:00
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    Moreover, there's an MO answer: https://mathoverflow.net/questions/118532/what-was-lamberts-solution-to-xmx-q , and the question has a link to the paper. – Chappers Jul 14 '17 at 17:06
  • @Chappers http://www.kuttaka.org/~JHL/L1758c.pdf this paper is in foreign language. Also the answer in mathoverflow does not include how the problem was solved. Thanks. – Amit Hasan Jul 14 '17 at 17:12
  • @Chappers the trinomial equation in the mathoverflow question is slightly different. – Amit Hasan Jul 14 '17 at 17:14
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    In general for solving $P(z) = a$ you'll need this formula for $P^{-1}$ – reuns Jul 14 '17 at 17:27

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Let use solve $x=q+x^{n+1}$ using power series in $q$. The idea is to use an iterative process with $x_0=q+O(q^2)$ and $x_{k+1}=q+x_k^{n+1}$. In the limit we get $$x = q + q^{n+1}+(n+1)x^{2n+1}+(n+1)(3n+2)/2x^{3n+1}+\cdots$$ where the coefficients are OEIS sequence A070914. In general, you could use the Lagrange inversion theorem. See the example in the Wikipedia article which also mentions Lambert W.

Somos
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