Taylor series of i[x]

[Xorter]

I'm not familiar with Sedenion, but it is an abstract algebra concept, not a complex analytic function, right?  So by definition, unless there is some mapping to a complex function, then it would not have a Taylor series...

Ah, yes, and I could recognise something interesting.
I tried to take i[x] into the Carleman matrix and to square it with itself, it seems successfully, maybe. Well, I got that the new Carleman matrix 2nd coloumn have I[1,0], I[1,1], I[1,2], ... where
\( I_{1,N} = 1/N! \sum_{k=1}^{\infty} i^{(N)k}_x i^{(k)}_x / k! |_{x=0} \)
So
\( i_x ^{o2} = i_{i_x} = \sum_{k=0}^{\infty} I_{1,k} x^k =
= \sum_{k=0}^{\infty} i^{(k)}_x / k! + x \sum_{k=0}^{\infty} i'^{k}_x i^{(k)}_x / k! + x^2 \sum_{k=0}^{\infty} i''^{k}_x i^{(k)}_x / k! + ... |_{x=0} \)

Now I feel we are closer than before.