A “half-iterate” of the Faulhaber-problem of “summing like powers”?

2018-04-07 08:08:37

Background: In a question about the "sum of sums of $k$th powers of first natural numbers" someone asked -in principle- for the 2'nd iteration of the Faulhaber-problem. The Faulhaber's problem can be stated as:

Find polynomials $f_p(n)$ in $n$ for each $p$ such that $f_p(n)$ represents equivalently

$$ S_p(n) = \sum_{k=1}^n k^p \tag 1$$

The second iteration is then assumed as to replace $k^p$ with the sums $S_p(k)$

$$ SS_p(n) = \sum_{k=1}^n S_p(k) \tag 2$$

In my answer I gave a solution, where I used the Faulhaber-polynomials organized in a matrix $G$ and that matrix been taken to the second power. The correctness of the solution can then nicely be seen using some cases of $n$ and the matrix to some handy size of, say $32 \times 32$ or so, which shows the results $SS_p(n)$ in the $p$th row of the result vector.

Current question: Having (2) now as proper second iteration I asked myself -just for curiousity- whether one can define also a "half-iterate" for this proble