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.

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    $\begingroup$ There are results on asymptotic behaviour (approaching 1 along the reals) for this type of function: D Handelman, Karamata Renewed and Local Limit Results Canad. J. Math. Vol. 58 (5), 2006 pp. 1026–1094, cms.math.ca/openaccess/cjm/v58/handelman3372.pdf, where the term used is Hadamard 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. $\endgroup$ Apr 19 '15 at 14:13
  • $\begingroup$ David Handelman pointed out that this is the Hadamard product of P with itself. Hadamard proved some results that might be useful, perhaps in J. Hadamard, Th´eoreme sur les s´eries entieres, 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. $\endgroup$ Apr 19 '15 at 19:47
  • $\begingroup$ Now I know the name for this! I looked at the papers and I think I can confirm what Timothy Chow was saying: namely that the degree of the denominator increases by a factor of $\kappa$ $\endgroup$ Apr 20 '15 at 2:15

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.


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