# Is there a generating function for any finite sequence of random numbers?

hypothesis: for any finite sequence of random positive integers exists a generating function.

If the hypothesis is true, is there a structured way of finding a generating function for a finite sequence of random positive integers?

Here is what I understand from the past 2 days of reading:

A generating function for Fibonacci numbers - eventhough it doesn't have much to do with my question - is: $$\begin{eqnarray*} \frac{1}{1− (x + x ^ 2)} \end{eqnarray*}$$ where: $$\begin{eqnarray*} \frac{1}{1− (x + x ^ 2)} =1 + x + 2 x ^ 2 + 3 x ^ 3 + 5 x ^ 4 + 8 x ^ 5+ \cdots \end{eqnarray*}$$ The coefficients are Fibonacci numbers, i.e. the sequence {1,1,2,3,5,8,13,21, ...}.

But if I have a finite sequence of random numbers, for example {4,5,0,9,1,5} Is there a method of finding its generating function?

• Cf. finite impulse response (FIR) filter. Apr 7, 2020 at 22:11
• Generating functions are defined for infinite sequences. Should we assume that your finite sequence continues with "0,0,0,..."? Or should we assume that there are further elements after the ones you have specified but they are unknown? Apr 8, 2020 at 9:29
• I'd recommend saying arbitrary instead of random if you don't talk about random processes at all. And if you do really mean random processes, be precise about distributions. Often people will say that a (perfect) die is "random", but a faked die with 30%-70% chances for head-tail is not random -- which is wrong! It's no longer uniformly random, that's true. Apr 8, 2020 at 12:09
• Reply to @FedericoPoloni: I always thought of a finite series as an infinite series that continues with "0,0,0,..." (as it relates to my problem). Would that still be possible if I gave you this information?
For a sequence with finitely many nonzero values, the corresponding generating function is just a polynomial. Your example has generating function $$4+5x+9x^3+x^4+5x^5$$.