# Primary reference for a formula related to the Faa di Bruno formula

In discussing Lie derivatives in his book Advanced Combinatorics (Reidel, 1974, pg. 220), Comtet refers to a formula of Pourchet related to the celebrated Faa di Bruno formula. Other references to Pourchet's formula that I've found always refer back to Comtet. I suspect that the actual author was Eugene Prouhet, but I have no access to his publications in the 1800s nor Lukacs paper on the FdB formula that might reference Prouhet.

(Edit 8/29/15 to clarify my interest: To whom was Comtet actually referring, i.e., to whom was Comtet attempting to attribute the formula in his publications? Ideally, I would like confirmation of the formula in a publication--primary source--by Prouhet or Pourchet.)

Does anyone know to whom Comtet was actually referring; i.e., is anyone aware of the primary source for Comtet's attribution?

(Edit after Conifold's initial answer: Note that I am not interested in the Faa di Bruno formula per se, just the primary source of Comtet's attribution.)

• You will find a reference in Number Theory and Related Fields: In Memory of Alf van der Poorten in this book Yves Pourchet is quoted if you go to google books you'll find the place. – Duchamp Gérard H. E. Aug 27 '15 at 9:00
• It seems that Pourchet (the very name in Comtet) was known and quoted by his community but did not publish much (a common french habit at that time). That's why Comtet says "The following result by Pourchet ..." but did not put him in the bibliography, I think. – Duchamp Gérard H. E. Aug 27 '15 at 9:02
• Here is more on Louis François Antoine Arbogast www-history.mcs.st-and.ac.uk/Biographies/Arbogast.html – Duchamp Gérard H. E. Aug 27 '15 at 13:55

Some preliminary remarks. Lukacs's paper does not refer to or mention Pourchet, Prouhet, or any similarly spelled name. Johnson gives a detailed history of Faa di Bruno's formula. He does not mention such a name either, but comments "one is struck by the obscurity of many of the names in our story... It would be nice to have more biographical information about some of them..."

I have a feeling that the only way to confirm would be to find Comtet's formula in Prouhet's papers. Comtet's book (p.331) references one Prouhet's paper in particular, in Nouvelles Annales M[athématiques], 5 (1866) 384, and nothing by "Pourchet". Unfortunately, Numdam shows no Prouhet's paper in 1866 Nouvelles Annales, on p.384 or elsewhere, and the one from 1867 has nothing of the sort in it. Then again Numdam does not show his 1864 and 1865 papers, which appear on Gallica-Math list. Both agree on nothing from Prouhet in 1866 though.

Even if it is there he may not have been the first to derive the formula. Faa di Bruno published 'his' formula in December of 1855, but "that several other mathematicians found different expressions for the m-th derivative of $g(f(t))$ in the 19th century has been forgotten; these are all independent of Faa di Bruno's work and a few of them predate it. Most of all, Faa di Bruno was neither the first to state the formula that bears his name, nor the first to prove it... Other mathematical accomplishments... have caused him to receive more credit for the higher chain rule than he deserves..."

The first proof appeared in volume 3 of Lacroix's Traite du Calcul, "a snapshot (or perhaps a mural) of 18th century analysis", 2nd edition only, which came out in 1819. His source appears to be 1800 Du Calcul des De'rivations of Arbogast, who "Arbogast gives a prose rule for writing down the general case, and on p. 313 he illustrates his rule by writing down the case n = 6 again, twice, followed by n = 7. Although he never gives a general formula, nor a proof of his rule... there is a formula for the coefficient of $x^m$ in $(\zeta + \gamma x + \delta x^2 + \varepsilon x^3 + \text{etc.})^n$. Thus Arbogast had most of the ingredients of Lacroix's argument at hand, but he seems never to have written down Faa di Bruno's formula as such.".

• Thanks for searching through Lukacs (+1). My search engine gave the hit but did not indicate the absence of Prouhet as it did for some related hits, hence my use of might. Johnson is the standard ref on the history, as most recent articles on FdB indicate. I scanned through some publications of Prouhet but didn't find any such formula--may be a red herring. Anyway, I'm curious if anyone has knowledge of the primary source of Comtet's attribution, not a comprehensive history of FdB. For others it's nice that you've given a brief aside on the general history. – Tom Copeland Aug 27 '15 at 2:33
• Interesting lead. Hope someone can check it out. – Tom Copeland Aug 27 '15 at 4:55
• Maybe the phrasing about Arbogast "gives a prose rule ... although, nor a proof of his rule" is a bit overshot or interpretated, have a look at the past section of (paragraph 3, Computation of the n-th derivative of a composite function) mathinfo.unistra.fr/fileadmin/upload/IREM/Publications/L_Ouvert/… "" – Duchamp Gérard H. E. Aug 27 '15 at 10:37
• "lack of depth of the subject" seems rather opinionated, or judgmental, kinda like saying the same for circumference/diameter, or $e^{ix}=cos(x)+i \; sin(x)$. Ubiquity would be a better term than shallow.. – Tom Copeland Aug 29 '15 at 22:04
• @Duchamp Gérard H. E. I inserted Johnson's full quote, it sounds more charitable. – Conifold Aug 31 '15 at 17:41