Geodesic
Defining sums of products of power sums
Journal of integer sequences, Tome 19 (2016) no. 1
We study the sums of products of power sums of positive integers and their generalizations, using the multiple products of their exponential generating functions. The generalizations include a closed form expression for the sums of products of infinite series of the form $\sum_{n=0}^{\infty}\alpha^n n^k, 0\vert\alpha\vert1, k\in\mathbb{N} _0$ and the related Abel sum, which define, in a unified way, the sums of products of the power sums for all integers $k$ and connect them with the zeta function.
Classification : 11A25, 11B68, 05A10, 11B65
Keywords: power sum of integers, Bernoulli number, Bernoulli polynomial, Riemann zeta function, exponential generating function, Apostol-Bernoulli number 
@article{JIS_2016__19_1_a7,
     author = {Singh,  Jitender},
     title = {Defining sums of products of power sums},
     journal = {Journal of integer sequences},
     year = {2016},
     volume = {19},
     number = {1},
     zbl = {1364.11013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a7/}
}
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Singh,  Jitender. Defining sums of products of power sums. Journal of integer sequences, Tome 19 (2016) no. 1. http://geodesic.mathdoc.fr/item/JIS_2016__19_1_a7/