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May 15th, 2011, 09:58 AM   #1
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a question about a power series

If we let be deltation, its commutative and associative, we find addition is distributive across it: , and that it is defined by the following recursion formula: , multiplication across deltation also behaves as exponentiation across addition.

So the question I had was, what's its taylor series?

I managed to work out the first five derivatives, I'm gonna leave out the steps, I just wonder if someone can maybe help me out with the algorithm or if they've seen these coefficients somewhere before (or maybe someone knows a technique for iterated differentiation):


Interestingly enough, the coefficients always add up to one.

The obvious choice is to center the taylor series about y, so if


therefore the taylor series will look something like this:

where etc...

So thanks for reading if you got this far, any help would be greatly appreciated.
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