We have a bit of a dilemma here. Though the ^n has advantages with respect to applying the \ and / notation, basicly its ambiguous with taking the n-th power.
f^n(x) = f(x)f(x)....f(x) or f^n(x)=f(f(...f(x)...)).
Of course you can say we can write the first one as f(x)^n, however then there is no operation to denote the n-th power of a function.
So we need a different symbol instead of ^. For example in maple there is this notation with \(n when taking the nth derivative with respect to x, you write diff(f,x\)n). So I propose - if we want to give up the <n> notation for the advantage of taking (\ and) / -
f\(n=\( f^{\circ n} \).
correspondingly
b[k]\)n(x), b/[k]\(n(x), etc as you already listed.
As now the \) is available because we have the notation [4] and [5] and dont need dedicated symbols like #, $ or ยง for them anymore.
f^n(x) = f(x)f(x)....f(x) or f^n(x)=f(f(...f(x)...)).
Of course you can say we can write the first one as f(x)^n, however then there is no operation to denote the n-th power of a function.
So we need a different symbol instead of ^. For example in maple there is this notation with \(n when taking the nth derivative with respect to x, you write diff(f,x\)n). So I propose - if we want to give up the <n> notation for the advantage of taking (\ and) / -
f\(n=\( f^{\circ n} \).
correspondingly
b[k]\)n(x), b/[k]\(n(x), etc as you already listed.
As now the \) is available because we have the notation [4] and [5] and dont need dedicated symbols like #, $ or ยง for them anymore.
Quote:Fortunately, however, we do not need a notation for auxiliary hyper-logarithms, because:Thats really clever and again reminds me on Szekeres consideration of the Abel function as an integral in "Scales of infinity and Abel's functional equation", 1984.
\( h
= \mathtt{b[N]\^{\backslash}z(x)}
= \left({}^N_b\begin{tabular}{|c} z \\\hline\end{tabular}\right)
- \left({}^N_b\begin{tabular}{|c} x \\\hline\end{tabular}\right)
\)
So if neccessary, this can be written \( h = \mathtt{b[N]{\backslash}z - b[N]{\backslash}x} \) which means we really don't need either my notation, nor GFR's notation for auxiliary hyper-logarithms.