06/01/2009, 06:14 PM
Let a be a base \( 1 < a < \e^{1/\e} \). Then we can build a regular tetration function \( tet_a \) around either of the fixed points. In either case, the function will be periodic, with period given by \( per(a) = 2 \pi \i / \log(\log(f)) \) for f the fixed point.
Thus the Laplace transform of sexp_a is:
\( tet_a(z) = \sum_{k \in Z} \e^{per(a) k z} c_k \)
Here, if we expand around the lower fixed point, all the positive coefficients will be zero, since the function tends to the fixed point at \( +\inf \). Similarly, if we expand around the upper fixed point, all the negative coefficients will be zero. In either case, \( c_0 \) is the chosen fixed point.
Now from the equation above, we have \( tet_a(z+1) = \sum_{k \in Z} \e^{per(a) k z} [\e^{per(a) k} c_k] \). But by definition, this is equal to \( \exp_a(tet_a(z)) = \sum_{n \in N} \frac{\(\sum_{k \in Z} e^{per(a) k z} c_k \log(a)\)^n}{n!} \).
By equating the terms of the resulting Laplace series, we get the equation \( c_k e^{per(a) k} = \sum_{n \in N} \frac{(\log a)^n}{n!} \sum_{\Sigma k_i = k} \[\prod_{i=1}^n c_{k_i}\] \). The inner sum is over all integer sequences of length n which sum to k. The finitude of this sum is ensured by the fact that either all positive or all negative coefficients are zero.
Thus the Laplace transform of sexp_a is:
\( tet_a(z) = \sum_{k \in Z} \e^{per(a) k z} c_k \)
Here, if we expand around the lower fixed point, all the positive coefficients will be zero, since the function tends to the fixed point at \( +\inf \). Similarly, if we expand around the upper fixed point, all the negative coefficients will be zero. In either case, \( c_0 \) is the chosen fixed point.
Now from the equation above, we have \( tet_a(z+1) = \sum_{k \in Z} \e^{per(a) k z} [\e^{per(a) k} c_k] \). But by definition, this is equal to \( \exp_a(tet_a(z)) = \sum_{n \in N} \frac{\(\sum_{k \in Z} e^{per(a) k z} c_k \log(a)\)^n}{n!} \).
By equating the terms of the resulting Laplace series, we get the equation \( c_k e^{per(a) k} = \sum_{n \in N} \frac{(\log a)^n}{n!} \sum_{\Sigma k_i = k} \[\prod_{i=1}^n c_{k_i}\] \). The inner sum is over all integer sequences of length n which sum to k. The finitude of this sum is ensured by the fact that either all positive or all negative coefficients are zero.