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Generating Series for Fibonacci Numbers

The generating (power) series for  Fibonacci numbers is

F(z) = F(0) + F(1) z + F(2) z^2 + ··· = Underscript[∑, n >= 0] F(n) z^n(1)

Adding  together
typeset structure
typeset structure
we get
typeset structure,
or in other words

F(z) - F(0) F(z) - F(0) - F(1) z FormBox[Cell[ ... z 2 z

which yields that

z FormBox[RowBox[{Cell[TextData[{F(z)=, ... 2 1 - z - z(2)

The polynomial typeset structure is called the characteristic polynomial for Fibonacci numbers.  Since typeset structure , it has two roots typeset structure, typeset structure.

The number typeset structure, called golden ratio . The second root  typeset structure.

Cite this web-page as:

Štefan Porubský: Generating Series for Fibonacci Numbers.

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