4
$\begingroup$

If I have a generating function for $f(n)$ defined by

$g(x)=\displaystyle\sum_{n=0}^{\infty}f(n)x^n=\dfrac{P(x)}{Q(x)}$,

where $P(x)$ and $Q(x)$ are polynomials and $Q(x)$ is not the zero function, how could I show that $f(n)$ is not more than exponential?

  • 0
    Got something from an answer below?2012-07-25

2 Answers 2

4
  1. Since $Q(0)\ne0$, $|Q(z)|\geqslant a$ for every $z$ in a neighborhood of $0$, say for every $|z|\leqslant\varepsilon$, for some $a\gt0$ and $\varepsilon\gt0$.
  2. Since $P$ is continuous, $|P(z)|\leqslant b$ for every $|z|\leqslant\varepsilon$, for some finite $b$.
  3. For every $n$, $ f(n)=\frac1{2\mathrm i\pi}\oint\frac{P(z)}{Q(z)}\frac{\mathrm dz}{z^{n+1}}, $ where the integral is over the circle of equation $|z|=\varepsilon$, whose length is $2\pi\varepsilon$.
  4. In particular, $ |f(n)|\leqslant\frac1{2\pi}2\pi\varepsilon\frac{b}a\frac1{\varepsilon^{n+1}}=cK^n, $ with $c=b/a$ and $K=1/\varepsilon$.
  • 0
    What you used is the Cauchy formula for the nth derivative of $\frac{P(x)}{Q(x)}$ at the point $x=0$ divide bu $n!$. It is the formula $ f(n)=\frac{1}{n!}{\left(\frac{P(0)}{Q(0)}\right)}^{(n)} = \frac{1} {2\pi i} \int_C \frac { P(z) }{ Q(z) }\frac{dz}{z^{n+1}} $.2012-08-11
0

If you look backward to your problem you see that $f(n)$ is nothing, but the nth derivative of your rational polynomial $\frac{P(x)}{Q(x)}$ divided by $ n! $. I refer you to my Ph.D thesis (section 6.2.1) which covers the topic of finding the nth derivative of any rational polynomial. In other words you get a closed form formula for the nth derivative which may help you to conclude you assertion:

https://docs.google.com/file/d/0B4FXAHVyGS9KMGRiNDMyNDctMmQ1NS00MDI3LTk2OWEtNzc3N2ZlNDVmYjJm/edit?hl=en_GB&pli=1

  • 0
    My point was to give him an idea to prove his assertion by exploiting the nth derivative o$f$ his rational polynomial.2012-07-17