While reading this paper, the author provides an alternative definition of the Lagrange inversion formula. Call me crazy, but my intuition tells me that there's something wrong with his derivation. Can anyone verify if his argument is correct or not?
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1$\begingroup$ See for example this mathoverflow.net/questions/145555/…. This method for finding the comp inverse can be applied to formal power series and allows formulations in different "coordinate/indeterminate" systems related to very interesting combinatorics. Very old method at least as far back as the first half of the 1800s. $\endgroup$– Tom CopelandCommented Jan 13, 2015 at 9:29
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$\begingroup$ This kind of expansion is called a Lie series. I didn't check the author's derivation, but formulas of this type are well known. Unfortunately I don't know of a really good introductory reference. $\endgroup$– Ira GesselCommented Jan 7, 2016 at 16:57
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$\begingroup$ @IraGessel, this might be a good intro to the historiy of the method, not sure if it includes anything about Abel though: books.google.com/…. I typed in the title and Graves to see a google books excerpt $\endgroup$– Tom CopelandCommented Jan 23, 2016 at 3:06
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$\begingroup$ A combinatorial approach to Lie series has been given by Gilbert Labelle in two papers: MR0814421 (87c:05007) Une combinatoire sous-jacente au théorème des fonctions implicites. [A combinatorial theory underlying the implicit function theorem] J. Combin. Theory Ser. A 40 (1985), no. 2, 377–393 and MR0787718 (86j:05015) Éclosions combinatoires appliquées à l'inversion multidimensionnelle des séries formelles. [Combinatorial bloomings applied to the multidimensional inversion of formal series] J. Combin. Theory Ser. A 39 (1985), no. 1, 52–82. $\endgroup$– Ira GesselCommented Jan 23, 2016 at 15:14
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