Pheww - that's a lot of jungle. Anyway: thanks for that collection of information.

I've not much to say: I second, concerning the differentation, that the Leibniz-notation has its advantages over the f'-notation. But this reminds me to another distinction: while in difficult problems, for instance where I'm searching for a solution, the Leibniz-notation helps sometimes, because you see even possibilities of cancelling, for a fluently written text the shorter one, f', is more convenient. So we have another demand, which leads possibly to different notations as well.

Also I would like to refer to one consideration of mine concerning the left-down subscript notation. Well - I didn't force it after I proposed it, but often, when I read articles now, I feel discomfortable with the notation a1^a2^a3^...^an - it is a bit misleading, since we evaluate it beginning from an and *not* from a1, so in fact it should-at least- be rewritten as an^...^a3^a2^a1. The left-down subscript would prevent this inconsistency inherently.

Also, for ASCII-notation, which is always needed, I preferred the {base,x}^^iteration - notation first; but to make it better fit into the usual binary-operator-scheme of most of our formulae, I tried with

x {operator,base}^iterator , which can then easily be concatenated and I feel is also a bit intuitive (possibly ° instead of ^ although ^ has already a certain common-understanding of iteration, but ° has much more). So x{^^,b}°h can be concatenated to

x {^^,b}°h {^^b}°k = x {^^,b}°(h+k)

and -in specialized texts- this may be extended to other operators

x {+,b}°h = x + b*h , x {*,b}°h = x * b^h , x{^,b}°h = x^b^h ,

x {^^,b}°h = b^b^b^...^x and for higher operators without specific symbols this can easily be extended to a general hierarchy

x {+,b}°h = x {o1,b}°h

x {*,b}°h = x {o2,b}°h

x {^,b}°h = x {o3,b}°h // if this is really needed

x {^^,b}°h = x {o4,b}°h

...

but - well, this adds again to the present jungle, and doesn't reflect the need for appropriate notation for inverse operations. So ...

The term "nested" is usually be taken for more complicated structures (like trees), and I would avoid it, if only "repeated" , "concatenated", "sequenced" is meant, like working through a linear list of sequential operations. Where I would use "repeated" if the same base is taken (as in tetration) and "concatenated" or "sequenced" or the like, if the bases are varying/undetermined.

So much for short, it's also a bit early in the morning...

Gottfried

I've not much to say: I second, concerning the differentation, that the Leibniz-notation has its advantages over the f'-notation. But this reminds me to another distinction: while in difficult problems, for instance where I'm searching for a solution, the Leibniz-notation helps sometimes, because you see even possibilities of cancelling, for a fluently written text the shorter one, f', is more convenient. So we have another demand, which leads possibly to different notations as well.

Also I would like to refer to one consideration of mine concerning the left-down subscript notation. Well - I didn't force it after I proposed it, but often, when I read articles now, I feel discomfortable with the notation a1^a2^a3^...^an - it is a bit misleading, since we evaluate it beginning from an and *not* from a1, so in fact it should-at least- be rewritten as an^...^a3^a2^a1. The left-down subscript would prevent this inconsistency inherently.

Also, for ASCII-notation, which is always needed, I preferred the {base,x}^^iteration - notation first; but to make it better fit into the usual binary-operator-scheme of most of our formulae, I tried with

x {operator,base}^iterator , which can then easily be concatenated and I feel is also a bit intuitive (possibly ° instead of ^ although ^ has already a certain common-understanding of iteration, but ° has much more). So x{^^,b}°h can be concatenated to

x {^^,b}°h {^^b}°k = x {^^,b}°(h+k)

and -in specialized texts- this may be extended to other operators

x {+,b}°h = x + b*h , x {*,b}°h = x * b^h , x{^,b}°h = x^b^h ,

x {^^,b}°h = b^b^b^...^x and for higher operators without specific symbols this can easily be extended to a general hierarchy

x {+,b}°h = x {o1,b}°h

x {*,b}°h = x {o2,b}°h

x {^,b}°h = x {o3,b}°h // if this is really needed

x {^^,b}°h = x {o4,b}°h

...

but - well, this adds again to the present jungle, and doesn't reflect the need for appropriate notation for inverse operations. So ...

The term "nested" is usually be taken for more complicated structures (like trees), and I would avoid it, if only "repeated" , "concatenated", "sequenced" is meant, like working through a linear list of sequential operations. Where I would use "repeated" if the same base is taken (as in tetration) and "concatenated" or "sequenced" or the like, if the bases are varying/undetermined.

So much for short, it's also a bit early in the morning...

Gottfried

Gottfried Helms, Kassel