Unique generating function for a sequence

Question

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 $F_0=0, F_1 = 1$?

Answer 1

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.