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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)], <differentiability prop.>, 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). - 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
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LINKS
| 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 (tcjpn(AT)msn.com), Oct 08 2008]
Tom Copeland, Mathemagical Forests v2 Posted June 2008
T. Copeland, Addendum to Mathemagical Forests
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