Crazy power series in terms of base e

[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.