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=.

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 § 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=.

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 § 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.

So if neccessary, this can be written which means we really don't need either my notation, nor GFR's notation for auxiliary hyper-logarithms.