Computing Abel function at a given center

[Gottfried]

To give a very basic example, we can recenter

\( f(x) = x^2 + 3x - 1 \)

to

\(
\begin{eqnarray}
g(x) & = & f(x+1) \\
& = & (x+1)^2 +3(x+1) - 1 \\
& = & x^2 + 5x + 5
\end{eqnarray}
\)

(...)

If we want to find f(1), we simply find g(0).
(...)

Ah, I see, it's as I expected. Then this is just the shift of
V(x)~ * Bs = V(y)~
to
V(x')~ * Qs = V(y')~
with x' = x/t-1 which I also perform by multiplication by the Pascal-matrix. Good to know.
It's the same way, as I described it occasionally, to get "exact terms", when the terms in the columns of the Bs-matrix are badly conditioned.
It is just

V(x)~ * Bs = V(y)~

where

Bs = dV(1/t)*P^-1~ * Q * P ~ * dV(t)

and then using associativity

V(x)~ * dV(1/t) * P^-1~ = V(x/t-1)~ = V(x')~

and then

V(x')~ * Q * P~ * dV(t) = V(y)~
V(x')~ * Q = V(y)~ *dV(1/t) * P^-1 ~
V(x')~ * Q = V(y/t-1)~ = V(y')~

where Q is triangular: for height=1 it contains the Stirling kind 2, and for height=-1 (the inverse) the stirling kind 1 numbers, factorial scaled, such that it performs for base b=exp(1)

V(x)~ * Q = V(exp(x)-1)~

or general base b

V(x)~ * Q = V(b^x-1)~

and its inverse (stirling kind 1)

V(x)~ * Q^-1 = V(log(1+x)/log(b)) ~


So essentially we do the same thing here, well...

Gottfried