Solving Linear Recurrences from Differential Equations in the Exponential Manner and Vice Versa

Abstract

While the ordinary generating function (OGF) for the Fibonacci-Sequence is well known,

$$ F\left( z \right) = \sum {{f_n}{z^n} = \frac{z}{{1 - z - {z^2}}}\left( {{\text{convergent for }}\left| z \right| < \frac{1}{2}\left( {\sqrt 5 - 1} \right)} \right)} $$

the exponential generating function (EGF)

$$ f\left( z \right) = \sum {{f_n}\frac{{{z^n}}}{{n!}} = \frac{1}{{\sqrt 5 }}\left( {{e^{\frac{1}{2}\left( {1 + \sqrt 5 } \right)z}} - {e^{\frac{1}{2}\left( {1 - \sqrt 5 } \right)z}}} \right)} \left( {z \in C} \right) $$

is not quite so common. But the latter representation is interesting with two respects: First, the well known Euler-Binet-formula is directly recognizable.