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I have the following function

$$f(x)=\frac{\sum_{n=0}^{+\infty}a_n \, x^{-n}}{\sum_{n=0}^{+\infty}\left(b_n \, x^n + c_n \, x^{-n}\right)}$$

Is there a way, maybe using long division to find the following equivalence:

$$\sum_{n=0}^{+\infty}d_n \, x^n=\frac{\sum_{n=0}^{+\infty}a_n \, x^{-n}}{\sum_{n=0}^{+\infty}\left(b_n \, x^n + c_n \, x^{-n}\right)}$$

or maybe with going from $-\infty$ to $\infty$, in order to find iteratively the values of $d_n$ ? I have to add that $a_n$, $b_n$ and $c_n$ are numerically known, but their explicit expression is unattainable. You can see that the biggest problem is that I have negative exponents at the numerator and the denominator.

Balfar
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  • An infinitude of values for $a_n$, $b_n$, and $c_n$ cannot be only numerically known, nor would long division be possible in a finite amount of time in that case. Are your summation bounds really $\infty$? – Andy Walls Apr 06 '23 at 10:04
  • Sine the RHS denominator can be written as $\sum_{n=-\infty}^\infty f_n x^n$, the summation on the LHS cannot possibly be limited to $n \ge 0$ in the general case. – Andy Walls Apr 06 '23 at 10:39

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