For an infinite sequence $\{a_n\}$, is the generating function unique to that sequence? Can I say for example that $\frac{x}{1-x-x^2}$ is the g.f. of the Fibonacci sequence $F_n$ with initial conditions $F_0=0, F_1 = 1$?

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    $\begingroup$ For every sequence, there are actually more than one generating function associated to it. The most common one is the OGF (ordinary generating function) followed by EGF (exponential generating function),... In general, if someone refer to g.f. of a sequence w/o other qualification, then one is referring to the OGF. $\endgroup$ – achille hui Apr 25 '19 at 8:50

Of course the generating function of a sequence $\{ a_n\}$ is unique, provided that you mean the same type of generating function. This is because the generating function of the sequence is a power series with coefficients uniquely determined by $a_n$. For example, the ordinary generating function is just $\sum a_nx^n$, the exponential generating function is $\sum(a_n/n!)x^n$, the Dirichlet series generating function is $\sum a_nn^{-s}$. You cannot therefore have two different generating functions (of the same type) corresponding to one sequence.

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