Crazy power series in terms of base e

[andydude]

The power series expansions of tetration \( {}^{z}a \) about the base are commonly discussed about (a=1), which produce very nice coefficients. However, I remembered that the same power series about other points made each coefficient depend on tetration, which isn't so bad if we already know one base. I'm not sure if this is useful for noninteger heights, because there are z's in the bounds for summation.

\(
\begin{tabular}{rl}
{}^{z}a
& = {}^{z}e \\
& +
\left[
\frac{1}{e} \sum_{k=1}^{z} \prod_{j=0}^{k} {}^{(z-j)}e
\right]
(a - e) \\
& +
\left[
\frac{1}{e^2}\sum_{k=1}^{z}
\left(
e(k-1) - 1 +
\sum_{j=0}^{k} \frac{1}{{}^{z-j}e}
\sum_{i=1}^{z-j}
\prod_{l=0}^{i} {}^{z-j-l}e
\right)
\prod_{j=0}^{k} {}^{z-j}e
\right]
(a - e)^2 \\
& + \cdots
\end{tabular}
\)

Pretty crazy...

Andrew Robbins

[Gottfried]

if we already know one base. I'm not sure if this is useful for noninteger heights, because there are z's in the bounds for summation.

Hi Andrew -

besides the rest of the formula (<phew>): there is an article of Markus Müller, Berlin, about sums with fractional bounds. Perhaps there is something interesting for you...

Gottfried


seeindex) : http://www.math.tu-berlin.de/~mueller/research.html


M. Müller, D. Schleicher, "Fractional Sums and Euler-like Identities", http://www.arxiv.org/abs/math/0502109

M. Müller, D. Schleicher, "How to add a non-integer number of terms, and how to produce unusual infinite summations", Journal of Computational and Applied Mathematics, Vol 178/1-2 pp 347-360 (2005), HowToAdd.pdf

[bo198214]

M. Müller, D. Schleicher, "Fractional Sums and Euler-like Identities", http://www.arxiv.org/abs/math/0502109

M. Müller, D. Schleicher, "How to add a non-integer number of terms, and how to produce unusual infinite summations", Journal of Computational and Applied Mathematics, Vol 178/1-2 pp 347-360 (2005), HowToAdd.pdf

Thank you Gottfried for the references.
The base idea was independently mentioned by Ansus on the forum:
http://math.eretrandre.org/tetrationforu...31#pid3131

For monomials we have a closed form for the (indefinite) sum.
Also we know that the sum is linear.
So we can compute the sum of polynomials and even powerseries.

[tommy1729]

M. Müller, D. Schleicher, "Fractional Sums and Euler-like Identities", http://www.arxiv.org/abs/math/0502109

M. Müller, D. Schleicher, "How to add a non-integer number of terms, and how to produce unusual infinite summations", Journal of Computational and Applied Mathematics, Vol 178/1-2 pp 347-360 (2005), HowToAdd.pdf

Thank you Gottfried for the references.
The base idea was independently mentioned by Ansus on the forum:
http://math.eretrandre.org/tetrationforu...31#pid3131

For monomials we have a closed form for the (indefinite) sum.
Also we know that the sum is linear.
So we can compute the sum of polynomials and even powerseries.

however , the non-integer number of additions in the pdf is just a possible generalisation.

so it might not be tetration afterall , or beter said , not neccessarily your " favorite " tetration.


regards

tommy1729