The following five mathematical objects, i.e.: a sequence, a generating function, a recurrence equation, an arithmetical function and an asymptotic estimate, are all representations of the well-know Fibonacci sequence.

$$(1,1,2,3,5,8,13,21,34,55,...)$$

$$g[x]=\frac{x}{1-x-x^2}$$

$$a[1]=1; a[2]=1; a[n]=a[n-1]+a[n-2]$$

$$F_n = \frac{1}{\sqrt{5}} (\frac{1+\sqrt{5}}{2})^n - \frac{1}{\sqrt{5}} (\frac{1-\sqrt{5}}{2})^n$$

$$f(x)=\frac{1}{\sqrt{5}}e^{x\log{\phi}}$$

So, if we have to solve a problem involving sequences we have various options to choose from to represent the sequence we are working with. I had quite a cognition after my first confront with the consequences of this concept. Summation for example. Summation of a sequence is equal to multiplication with $\frac{x}{1-x}$ in the generating function realm. The GF of $(1,1,1, \cdots )$ is $\frac{1}{1-x}$ which means that the GF of $(1,2,3, \cdots )$ must be $\frac{x}{(1-x)^2}$.

1-2017 More on the randomness of randomness.

2 months ago

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