|
|
COMMENTS
|
For more detail, including connections to Legendre transformations, rooted trees, A139605, A139002 and A074060, see Copeland link pg. 9.
For connections to the h-polynomials associated to the refined f-polynomials of permutohedra see my comments in A008292 and A049019.
Contribution by Tom Copeland Oct 14 2011: (Start)
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,x)= exp[s g(x)d/dx]x = F[s+FI(x)] <flow fct.>,
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
W[s,W(r,x)] = F{s+FI[F[r+FI(x)]]} = F[s+r+FI(x)] = W(s+r,x) <group property>. (See MF link below.)(End)
dW(s,x)/ds - g(x)dW(s,x)/dx = 0, so (1,-g(x)) are the components of a vector orthogonal to the gradient of W and, therefore, tangent to the contour of W, at (s,x) <tangency property>. - Tom Copeland. Oct 26 2011
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) eval. 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
Related to the logarithm of the formal group law for elliptic curves.- Tom Copeland, Feb. 24 2012
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, April 4 2012
|
|
|
LINKS
|
Table of n, a(n) for n=0..18.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 .
Tom Copeland, Flipping Functions with Permutohedra Posted Oct. 2008 [From Tom Copeland, Oct 08 2008]
Tom Copeland, Mathemagical Forests v2 Posted June 2008
T. Copeland, Addendum to Mathemagical Forests
V. Goryainov and O. Kudryavtseva One-parameter semigroups of analytic functions, fixed points and the Koenigs function Sbornik: Mathematics, 202:7 (2011), 971-1000
J. Hoffman, Topics in Elliptic Curves and Modular Forms pg. 10
C. Houzel, The Work of Niels Henrik Abel The Legacy of Niels Henrik Abel-The Abel Bicentennial, Oslo 2002 (Editors O. Laudal and R Piene), Springer-Verlag (2004), pg. 24-25
Wikipedia, Abel equation
|
