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# Formelsammlung Mathematik: Erzeugende Funktionen

• ${\displaystyle G\{f_{n}\}}$: Erzeugende Funktion der Folge ${\displaystyle f_{n}}$
• ${\displaystyle {\mathcal {Z}}\{f_{n}\}}$: Z-Transformierte (unilateral) der Folge ${\displaystyle f_{n}}$
• Konvergenzbereich: für ${\displaystyle x\in \mathbb {C} }$
${\displaystyle f_{0},f_{1},f_{2},f_{3},f_{4},\ldots }$ ${\displaystyle f_{n}}$ ${\displaystyle G\{f_{n}\}(x)}$ ${\displaystyle {\mathcal {Z}}\{f_{n}\}(z)}$ Konvergenzbereich
1, 0, 0, 0, 0, … ${\displaystyle [n=0]}$ ${\displaystyle 1}$ ${\displaystyle 1}$ ${\displaystyle \mathbb {C} }$
0, 1, 0, 0, 0, … ${\displaystyle [n=1]}$ ${\displaystyle x}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle \mathbb {C} }$
0, 0, …, 0, 1, 0, … ${\displaystyle [n=k]}$ ${\displaystyle x^{k}}$ ${\displaystyle {\Big (}{\frac {1}{z}}{\Big )}^{k}}$ ${\displaystyle \mathbb {C} }$
1, 1, 1, 1, 1, … ${\displaystyle 1}$ ${\displaystyle {\frac {1}{1-x}}}$ ${\displaystyle {\frac {z}{z-1}}}$ ${\displaystyle |x|<1}$
1, q, q2, q3, q4, … ${\displaystyle q^{n}}$ ${\displaystyle {\frac {1}{1-qx}}}$ ${\displaystyle {\frac {z}{z-q}}}$ ${\displaystyle |qx|<1}$
1, −1, 1, −1, 1, … ${\displaystyle (-1)^{n}}$ ${\displaystyle {\frac {1}{1+x}}}$ ${\displaystyle {\frac {z}{z+1}}}$ ${\displaystyle |x|<1}$
0, 1, 2, 3, 4, … ${\displaystyle n}$ ${\displaystyle {\frac {x}{(1-x)^{2}}}}$ ${\displaystyle {\frac {z}{(z-1)^{2}}}}$ ${\displaystyle |x|<1}$
1, 2, 3, 4, 5, … ${\displaystyle n+1}$ ${\displaystyle {\frac {1}{(1-x)^{2}}}}$ ${\displaystyle {\frac {z^{2}}{(z-1)^{2}}}}$ ${\displaystyle |x|<1}$
a, 1+a, 2+a, 3+a, … ${\displaystyle n+a}$ ${\displaystyle {\frac {a+(1-a)x}{(1-x)^{2}}}}$ ${\displaystyle {\frac {az^{2}+(1-a)z}{(z-1)^{2}}}}$ ${\displaystyle |x|<1}$
0, 1, 4, 9, 16, … ${\displaystyle n^{2}}$ ${\displaystyle {\frac {x(x+1)}{(1-x)^{3}}}}$ ${\displaystyle {\frac {z(z+1)}{(z-1)^{3}}}}$ ${\displaystyle |x|<1}$
1, 0, 1, 0, 1, … ${\displaystyle [2\,|\,n]}$ ${\displaystyle {\frac {1}{1-x^{2}}}}$ ${\displaystyle {\frac {z^{2}}{z^{2}-1}}}$ ${\displaystyle |x|<1}$
0, 1, 0, 1, 0, … ${\displaystyle [2\,|\,n+1]}$ ${\displaystyle {\frac {x}{1-x^{2}}}}$ ${\displaystyle {\frac {z}{z^{2}-1}}}$ ${\displaystyle |x|<1}$