Revision History for A145271
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| A145271
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Coefficients for expansion of (g(x)d/dx)^n g(x); refined Eulerian numbers for calculating compositional inverse of h(x) = (d/dx)^(-1) 1/g(x); iterated derivatives as infinitesimal generators of flows.
(history;
published version)
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#261 by R. J. Mathar at Tue Aug 06 10:08:24 EDT 2024
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#260 by R. J. Mathar at Tue Aug 06 10:07:54 EDT 2024
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C. Brouder, <a href="httphttps://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.180.7535&rep=rep1&type=pdf/a51b88800479e028b0f174aadbeb81dd643990be">Trees, renormalization, and differential equations</a>, BIT Numerical Mathematics, 44: 425-438, 2004, (Flow / autonomous differential equation, p. 429).
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approved
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#259 by N. J. A. Sloane at Tue Nov 22 22:55:41 EST 2022
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#258 by Jon E. Schoenfield at Sat Oct 08 22:32:56 EDT 2022
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#257 by Jon E. Schoenfield at Sat Oct 08 22:32:51 EDT 2022
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| COMMENTS
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Given analytic functions F(x) and FI(x) such that F(FI(x))=FI(F(x))=x about 0, i.e., they are compositional inverses of each other, then, with g(x) = 1/dFI(x)/dx, a flow function W(s,x) can be defined with the following relations:
W(s,0) = F(s), <orbit of the flow>,
W(0,x) = x <identity property>,
dW(0,x)/ds = g(x) = F'[FI(x)], <infinitesimal generator>, implying
dW(0,F(x))/ds = g(F(x)) = F'(x), <autonomous diff. eqn.>, and
Though A139605 contains A145271, the op. of A145271 contains that of A139605 in the sense that exp(s g(x)d/dx) w(x) = w(F(s+FI(x))) = exp((exp(s g(x)d/dx)x)d/du)w(u) evaluated at u=0. This is reflected in the fact that the forest of rooted trees assoc. to (g(x)d/dx)^n, FOR_n, can be generated by removing the single trunk of the planted rooted trees of FOR_(n+1). - _Tom Copeland_, Nov 29 2011
= exp((exp(s g(x)d/dx)x)d/du)w(u) evaluated at u=0. This is reflected in the fact that the forest of rooted trees assoc. to (g(x)d/dx)^n, FOR_n, can be generated by removing the single trunk of the planted rooted trees of FOR_(n+1). - Tom Copeland, Nov 29 2011
The functional equation W(s,x) = F(s+FI(x)), or a restriction of it, is sometimes called the Abel equation or Abel's functional equation (see Houzel and Wikipedia) and is related to Schröder's functional equation and Koenigs functions for compositional iterates (Alexander, Goryainov and Kudryavtseva). - Tom Copeland, Apr 04 2012
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Luc Rousseau, <a href="/A145271/b145271.txt">Table of n, a(n) for n = 0..9295</a> (Rows rows 0 to 25, flattened).
Wikipedia, <a href="https://en.wikipedia.org/wiki/Renormalization_group#Exact_renormalization_group_equations">Renormaliztion Renormalization Group</a>
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| FORMULA
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where (j')^k = ((d/dx)^j g(x))^k. And R^(n-1) g(x) evaluated at x=0 is the n-th Taylor series coefficient of the compositional inverse of h(x) = (d/dx)^(-1) 1/g(x), with the integral from 0 to x.
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| EXAMPLE
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Let h(x) = log((1+a*x)/(1+b*x))/(a-b); then, g(x) = 1/(dh(x)/dx) = (1+ax)(1+bx), so (0')=1, (1')= a+b, (2')= 2ab, evaluated at x=0, and higher order derivatives of g(x) vanish. Therefore, evaluated at x=0,
R^3 g(x) = (a+b)^3 + 4 *(a+b)*2ab = a^3 + 11 a^2*b + 11 ab^2 + b^3
R^4 g(x) = (a+b)^4 + 11 *(a+b)^2*2ab + 4 *(2ab)^2
etc., and these bivariate Eulerian polynomials (A008292) are the first few coefficients of h^(-1)(x) = (e^(ax) - e^(bx))/(a*e^(bx) - b*e^(ax)), the inverse of h(x). (End)
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#256 by Tom Copeland at Sat Oct 08 19:58:29 EDT 2022
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#255 by Tom Copeland at Sat Oct 08 19:58:21 EDT 2022
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Tom Copeland, <a href="https://tcjpn.wordpress.com/2022/10/08/compilation-of-oeis-partition-polynomials-a133314-a134685-a145271-a356144-and-a356145/">Compilation of OEIS Partition Polynomials A133314, A134685, A145271, A356144, and A356145</a>, 2022.
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#254 by Tom Copeland at Thu Oct 06 16:20:18 EDT 2022
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#253 by Tom Copeland at Thu Oct 06 16:19:50 EDT 2022
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| FORMULA
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Indeterminate substitutions as illustrated in A356145 lead to [E] = [L][P] = [P][E]^(-1)[P] = [P][RT] and [E]^(-1) = [P][L] = [P][E][P] = [RT][P], where [E] contains the refined Eulerian partition polynomials of this entry; [E]^(-1), A356145, the inverse set to [E]; [P], the permutahedra polynomials of A133314; [L], the classic Lagrange inversion polynomials of A134685; and [RT], the reciprocal tangent polynomials of A356144. Since [L]^2 = [P]^2 = [RT]^2 = [I], the substitutional identity, [L] = [E][P] = [P][E]^(-1) = [RT][P], [RT] = [E]^(-1)[P] = [P][L][P] = [P][E], and [P] = [L][E] = [E][RT] = [E]^(-1)[L] = [RT][E]^(-1). - _Tom Copeland_, Oct 05 2022
[E]^(-1) = [P][L] = [P][E][P] where [E] contains the refined Eulerian partition polynomials of this entry; [E]^(-1), A356145, the inverse set to [E]; [P], the permutahedra polynomials of A133314; and [L], the classic Lagrange inversion polynomials of A134685. Since [L]^2 = [P]^2 = [I], the substitutional identity, [L] = [E][P] and [P] = [L][E]. Tom Copeland, Oct 05 2022
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| CROSSREFS
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Cf. A133314, A036039, A133314, A356144, A356145.
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#252 by Tom Copeland at Wed Oct 05 12:59:14 EDT 2022
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