Tetra-series

[andydude]

If F is all zeroes except for a one somewhere, then that represents an \( x^n \) function. In general for integer n, the G's look like this

\(
\begin{tabular}{rl}
\frac{1}{x^4} &= 5 - 14x + 35({}^{2}x) - \frac{245}{3}({}^{3}x) + \frac{1957}{12}({}^{4}x) + \cdots \\
\frac{1}{x^3} &= 4 - 9x + 19({}^{2}x) - 39({}^{3}x) - \frac{817}{12}({}^{4}x) + \cdots \\
\frac{1}{x^2} &= 3 - 5x + \frac{17}{2}({}^{2}x) - 15({}^{3}x) - \frac{533}{24}({}^{4}x) + \cdots \\
\frac{1}{x} &= 2 - 2x + \frac{5}{2}({}^{2}x) - \frac{11}{3}({}^{3}x) + \frac{35}{8}({}^{4}x) + \cdots \\
1 &= 1({}^{0}x) + 0 \\
x &= 0 + 1({}^{1}x) \\
x^2 &= -1 + x + \frac{3}{2}({}^{2}x) - \frac{2}{3}({}^{3}x) - \frac{5}{24}({}^{4}x) + \cdots \\
x^3 &= -2 + \frac{7}{2}({}^{2}x) - \frac{41}{24}({}^{4}x) + \frac{37}{20}({}^{5}x) + \cdots \\
x^4 &= -3 - 2x + 5({}^{2}x) + 3({}^{3}x) - \frac{37}{12}({}^{4}x) + \cdots
\end{tabular}
\)

The first coefficient seems to have a pattern in it, but this is just because \( g_0 = f(1) - f'(1) = 1 - n \).

Oh, and another weird thing: \( {}^{\infty}x = 0 + {}^{n}x \) when approximated in this way.