Suppose you are given a power series $P=\sum_{i=0}^\infty{a_nt^n}$. I am primarily concerned with those power series coming from rational functions of the form $$ \frac{1}{\prod_{i=1}^k{(1-t^{\alpha_i})}}.$$ My motivation comes from looking at Hilbert series of noetherian freely generated subrings of polynomial rings. In general, I know $P$ and its expression as a rational function. I then consider the power series $P'=\sum_{i=0}^\infty{a_n^\kappa t^n}$ for some $\kappa\in\mathbb{Z}_{>0}$. I am interested in determining facts about the rational function that describes this power series (there should be one as it is a Hilbert series of a ring). Of greatest interest to me is the degree of the polynomial appearing in the denominator.

It's a general fact about any rational power series

$$\sum_{n=0}^\infty a_n t^n = {p(t) \over \prod_{i=1}^r (1-\gamma_i t)^{d_i}}$$

(where $p(t)$ is some polynomial of degree less than $\sum_{i=1}^r d_i$ and the $\gamma_i$ are distinct) that

$$a_n = \sum_{i=1}^r p_i(n)\gamma_i^n$$

for some polynomials $p_i$ of degree less than $d_i$. Conversely, any $a_n$ that can be expressed this way are the coefficients of a rational power series. See for example Theorem 4.1.1 in Richard Stanley's book *Enumerative Combinatorics*. From this it is immediate that $a_n^\kappa$ are the coefficients of a rational power series, as you say, and moreover by chasing through the proof of the theorem you can construct the denominator that you are interested in. I think that this shows that the degree will go up by at most a factor of $\kappa$, but to compute the degree exactly, you may have to keep careful track of which roots of unity occur with what multiplicities.

Karamata Renewed and Local Limit ResultsCanad. J. Math. Vol. 58 (5), 2006 pp. 1026–1094, cms.math.ca/openaccess/cjm/v58/handelman3372.pdf, where the term used isHadamard power, pp 24--25 (and Hadamard products are also discussed); this may be helpful. This applies to a special family of analytic functions, although among other things it requires nicely behaved nonnegative entries in the Maclaurin expansion. $\endgroup$`eme sur les s´eries enti`

eres, Acta. Math. 22 (1899) 55–63. I haven't tracked down that reference yet, though papers that cite it mention some theorems that might be useful. $\endgroup$