# Looking for references for an implicit differentiation formula

In a paper which I submitted to a peer-reviewed math journal in April 2010, I proved a formula for the n-th derivative $\frac{d^n z}{dw^n}$ in terms of (as a polynomial over the integers) of the partial derivatives of a given implicit function, $G(z,w)=0$, with respect to $z$ and $w$ (and negative integer powers of the "separant", $G_z$, the first partial derivative of $G$ with respect to $z$).

This is classic first-semester calculus homework exercise: to compute $\frac{d^n z}{dw^n}$ for n=1 and 2, namely,

$\frac{dz}{dw} = - \frac{G_w}{G_z}$

$\frac{d^2 z}{dw^2} = - \frac{G_{zz}G_{ww}}{G_{zzz}} + 2\frac{G_{zw}G_w}{G_{zz}} - \frac{G_{ww}}{G_z}$

I did so not knowing whether any one had proved the general formula first, because I am busy building on, generalizing, and using this result for other things, including chemical processing.

I have since proved the partial differential generalization of this implicit differentiation formula: i.e. given $G(z,w_1,...,w_N)=0$, compute

$\frac{d^{({u_1}+...+u_N})}{{dw_1}^{u_1}\cdots {dw_N}^{u_N}}z$ as a Laurent polynomial over the integers of the partial derivatives of $G$ with respect to $w_1,...,w_N$, and $z$

I do not have access to most peer-reviewed journals. I have had to make do with Google searches, Wolfram Research's MathWorld, online searches through my county library, and the help from one mathematician friend who has sent me related papers.

Most of the papers my friend sent me concern the Faa da Bruno formula (FdBF) and its generalizations, and the Lagrange Inversion Formula (LIF). Both the FdBF and LIF are very closely related to what I am doing, but they can not be trivially applied to get (my) general formula. (I tried... for about 8 months.) I have studied G.P. Egorychev's book: "Integral Representations of Combinatorial Sums" intensely, especially the back, with the multivariable generalizations of the LIF.

No, I am not in school. This is not a homework problem. This is "free-lance" research. I am not asking for a solution to the problem (as I already solved it independently).

I simply want to know yes or no whether someone has already done this. And, if so, where.

Thank you for the responses, in particular to go to arxiv.org, which I forgot, since I had submitted 2 papers there myself.

• if you sent the paper to a reasonable journal, there is some referee somwhere now who is working to answer precisely to this question, so just be patient – Piero D'Ancona Aug 26 '10 at 2:53
• I don't have a particular reference, but I think that your formula would not interest research mathematics journals. If you have a nice exposition of it, you might interest journals that mix research and expository work. Alternately, if your coefficients count some interesting combinatorics problem, it's possibly of interest in combinatorics --- I don't know that literature well enough to judge. (continued) – Theo Johnson-Freyd Aug 26 '10 at 2:55
• (continuation) But it's maybe not completely well known by mathematicians, but it's certainly known, how to compute the d^(u(1)+...+u(N)) z/ dw(1)^u(1) * ... * dw(N)^u(N) in terms of derivatives of G, at least for fixed 1+...+N. I should say also that I know absolutely nothing about chemical processing and related literatures --- if you have applications, I'm sure that a "mathematical methods in chemical processing" journal might be interested. – Theo Johnson-Freyd Aug 26 '10 at 2:57
• You do not seem to mention the arXiv: arxiv.org  The above is one version of the front page. In the upper right is a window for ordinary search, or in smaller type "Advanced search"  arxiv.org/find  which allows you to specify up to three categories, each out of twelve fields such as Author, Title, Abstract, Subject Description. I found six papers with Lagrange Inversion in the title.  The arXiv is not as flexible as MathSciNet and a percentage of papers are simply not legitimate. But it is becoming common for researchers to post there first. – Will Jagy Aug 26 '10 at 3:17
• I'd like to second Jagy's comments. Put your paper on the arXiv and then seek out comments on the content of your paper. That should give you a sense for where to submit your paper for publication. Just a note, I tried to TeXify your question to make it a little easier on the eyes. I hope I didn't mess anything up! – Ryan Budney Aug 26 '10 at 3:35

Zentralblatt has a sample service where you get 3 responses. I think you might want to look at the first item:

Zbl 1186.92069 Pongor, Gábor; Eöri, János; Rohonczy, János; Kolos, Zsuzsanna Direct inversion in the spectral subspace: a novel method for quantitative and qualitative analysis of chemical mixtures. (English) J. Math. Chem. 47, No. 3, 1085-1105 (2010). MSC2000: *92E99 65F15 92E10

Zbl pre05669072 Wang, Weiping Generalized higher order Bernoulli number pairs and generalized Stirling number pairs. (English) J. Math. Anal. Appl. 364, No. 1, 255-274 (2010). MSC2000: *11-99 05-99

Zbl 1183.74370 Wang, Meng-Fu; Au, F.T.K. Precise integration methods based on Lagrange piecewise interpolation polynomials. (English) Int. J. Numer. Methods Eng. 77, No. 7, 998-1014 (2009). MSC2000: *74S30 65D30 

 Zbl 1186.92069 Pongor, Gábor; Eöri, János; Rohonczy, János; Kolos, Zsuzsanna Direct inversion in the spectral subspace: a novel method for quantitative and qualitative analysis of chemical mixtures. (English) [J] J. Math. Chem. 47, No. 3, 1085-1105 (2010). ISSN 0259-9791; ISSN 1572-8897

Summary: A novel method, called Direct Inversion in the Spectral Subspace (DISS), has been developed for the quantitative (and partly qualitative) analysis of chemical mixtures. The method belongs to the broad group of supervised classification'' methods: its use necessitates the components'pure'' spectra, either experimental or computed. On the basis of three simple conditions, an elegant, linearized system of equations has been deduced, taking into account a sole restriction via the Lagrange multiplier method. This restriction is seemingly redundant but it has been shown that with its use the unknown normalization constant of the components' descriptive weighted average (CDWA) spectrum can be taken into consideration. The system of linearized equations can be solved repeatedly until convergence. Any kind of spectra can be used; the method does not require the non-negativity of spectral data points. Two versions of the new method have been developed: the normalized and the non-normalized versions regarding the components' spectra. In ideal cases, the non-normalized version of the DISS method provides a mixture's accurate composition due to the iteration for getting the correct norm of the CDWA spectrum. Realistically, the normalized version of the DISS method identifies a mixture's composition within a few molar percentage points accuracy, according to the test results in IR and $^{1}$H-NMR spectroscopy. The normalized method functions without any calibration measurements and needs only a control of accuracy; it is hoped that it will be a useful tool for chemical and biochemical analysis as well as for spectral databases. The DISS method is also useful for qualitative analyses in a limited sense: in the case of computed spectra of the components the set of the de facto components determined could be somewhat wider than those existing in the real system.

MSC 2000: *92E99 Appl. of mathematics to chemistry 65F15 Eigenvalues (numerical linear algebra) 92E10 Molecular structures

Keywords: decomposition of molecular spectra; Lagrange multiplier; quantitative analysis; qualitative analysis; IR; NMR; EPR; UV/Vis; Raman; CD; VCD; hexa-chloro-buta-1,3-diene; dioxane; D-Camphor; L-Menthol; supervised classification; spectral databases

• The keywords are very impressive! – Wadim Zudilin Oct 11 '10 at 11:14