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Friday, March 4, 2011

Different representations of sequences

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 $\frac{x}{(1-x)^2}$.

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Mathematics: is it the fabric of MEST?
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(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)