We know that we can express the Abel function \( \alpha \) of \( f \) at \( a \) by
\( \alpha = \log_c \circ \chi \) where \( \chi \) is the Schröder function of \( f \) at \( a \), where \( c=f'(a) \).
This means that the inverse of the Abel function which is actually the superexponential can be expressed by
\( F(x)=\chi^{-1}(c^x) \) with appropriate translation along the x-axis choosen such that \( F(0)=1 \). Here \( c=\exp_b'(a)=\ln(b) \exp_b(a)=\ln(b)a=\ln(b^a)=\ln(a) \)
The coefficients of \( \eta+a=\chi^{-1} \), \( \eta_0=0 \), \( \eta_1=1 \), can be recursively computed from the equation
(*) \( \eta(cx)=f(\eta(x)) \), \( f(x)=b^{x+a}-a = a (b^x-1) \)
by the composition formula
\( (f\circ g)_n = \sum_{k=1}^n f_k (g^k)_n \) where \( (g^k)_n = \sum_{n_1+\dots+n_k = n} f_{n_1} \dots f_{n_k} \).
Now we put formula (*) in:
\( c^n \eta_n = \sum_{k=1}^n f_k (\eta^k)_n \)
On the right side \( \eta_n \) only occurs for \( k=1 \) in \( \eta^k \), namely in the summand \( f_1 (\eta^1)_n = c \eta_n \). Thatswhy we have the recursive formula:
\( \eta_n = \frac{1}{c^n - c} \sum_{k=2}^n f_k (\eta^k)_n \)
So each coefficient of \( \eta \) is a polynomial in \( \ln(b) \) (\( b \) base) and a rational function in \( c=\ln(a) \) (\( a \) fixed point).
and upto translation along the x-Axis we have \( {^xb}=\eta(c^x)+a \).
\( \alpha = \log_c \circ \chi \) where \( \chi \) is the Schröder function of \( f \) at \( a \), where \( c=f'(a) \).
This means that the inverse of the Abel function which is actually the superexponential can be expressed by
\( F(x)=\chi^{-1}(c^x) \) with appropriate translation along the x-axis choosen such that \( F(0)=1 \). Here \( c=\exp_b'(a)=\ln(b) \exp_b(a)=\ln(b)a=\ln(b^a)=\ln(a) \)
The coefficients of \( \eta+a=\chi^{-1} \), \( \eta_0=0 \), \( \eta_1=1 \), can be recursively computed from the equation
(*) \( \eta(cx)=f(\eta(x)) \), \( f(x)=b^{x+a}-a = a (b^x-1) \)
by the composition formula
\( (f\circ g)_n = \sum_{k=1}^n f_k (g^k)_n \) where \( (g^k)_n = \sum_{n_1+\dots+n_k = n} f_{n_1} \dots f_{n_k} \).
Now we put formula (*) in:
\( c^n \eta_n = \sum_{k=1}^n f_k (\eta^k)_n \)
On the right side \( \eta_n \) only occurs for \( k=1 \) in \( \eta^k \), namely in the summand \( f_1 (\eta^1)_n = c \eta_n \). Thatswhy we have the recursive formula:
\( \eta_n = \frac{1}{c^n - c} \sum_{k=2}^n f_k (\eta^k)_n \)
So each coefficient of \( \eta \) is a polynomial in \( \ln(b) \) (\( b \) base) and a rational function in \( c=\ln(a) \) (\( a \) fixed point).
and upto translation along the x-Axis we have \( {^xb}=\eta(c^x)+a \).