Negate, Augment, Denominate

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NAD.  Note 1

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This old puzzle continues to come up in various contexts of discussion.
It's been too long and I can no longer remember what it had to do with
modular forms, but I wnated to keep thinking about it, so until I work
up the quantum to go look it up, I'll break it out for working on here.

Let C be the complex plane.
Let R be the real line.
Let * be the Kleene star.

Define A : C -> C such that A : x ~> x+1.
Define D : C -> C such that D : x ~> 1/x.
Define N : C -> C such that N : x ~> - x.

By way of mnemonics:
"A" is for Augment,
"D" is for Denominate,
"N" is for Negate.

For some reason that I can't remember either,
I used to write these operators on the right.

For example, pick a number on the real line,
let us say, x = 3.  Then we have this data:

xA = 4
xD = 1/3 = 0.3*
xN = -3

Among the important compound operators
are several trigrams like NAD, DNA, etc.

For example:

xNAD = 1/(1-x)

Formally speaking, one's formal knee jerks to give this:

xNAD = 1 + x + x^2 + x^3 + ...

But!  The second equation holds
only when the series converges.

For example:

(1/3)NAD = 1/(1-(1/3)) = 1/(2/3) = 3/2,

= 1 + (1/3) + (1/3)^2 + (1/3)^3 + ...

But:

3NAD = 1/(1-3) = -1/2

=/= 1 + 3 + 3^2 + 3^3 + ...

Anyway, I hope this is right -- it's been a while.

Exercise for the reader:

What can you say in general about x(NAD)^k
for k = 1, 2, 3, ...?

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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