Formelsammlung Mathematik: Erzeugende Funktionen

↑ Formelsammlung Mathematik

$f_{0},f_{1},f_{2},f_{3},f_{4},\ldots$	$f_{n}$	$G\{f_{n}\}(x)$	${\mathcal {Z}}\{f_{n}\}(z)$	Konvergenzbereich
1, 0, 0, 0, 0, …	$[n=0]$	$1$	$1$	$\mathbb {C}$
0, 1, 0, 0, 0, …	$[n=1]$	$x$	${\frac {1}{z}}$	$\mathbb {C}$
0, 0, …, 0, 1, 0, …	$[n=k]$	$x^{k}$	${\Big (}{\frac {1}{z}}{\Big )}^{k}$	$\mathbb {C}$
1, 1, 1, 1, 1, …	$1$	${\frac {1}{1-x}}$	${\frac {z}{z-1}}$	$\|x\|<1$
1, `q`, `q`², `q`³, `q`⁴, …	$q^{n}$	${\frac {1}{1-qx}}$	${\frac {z}{z-q}}$	$\|qx\|<1$
1, −1, 1, −1, 1, …	$(-1)^{n}$	${\frac {1}{1+x}}$	${\frac {z}{z+1}}$	$\|x\|<1$
0, 1, 2, 3, 4, …	$n$	${\frac {x}{(1-x)^{2}}}$	${\frac {z}{(z-1)^{2}}}$	$\|x\|<1$
1, 2, 3, 4, 5, …	$n+1$	${\frac {1}{(1-x)^{2}}}$	${\frac {z^{2}}{(z-1)^{2}}}$	$\|x\|<1$
`a`, 1+`a`, 2+`a`, 3+`a`, …	$n+a$	${\frac {a+(1-a)x}{(1-x)^{2}}}$	${\frac {az^{2}+(1-a)z}{(z-1)^{2}}}$	$\|x\|<1$
0, 1, 4, 9, 16, …	$n^{2}$	${\frac {x(x+1)}{(1-x)^{3}}}$	${\frac {z(z+1)}{(z-1)^{3}}}$	$\|x\|<1$
1, 0, 1, 0, 1, …	$[2\,\|\,n]$	${\frac {1}{1-x^{2}}}$	${\frac {z^{2}}{z^{2}-1}}$	$\|x\|<1$
0, 1, 0, 1, 0, …	$[2\,\|\,n+1]$	${\frac {x}{1-x^{2}}}$	${\frac {z}{z^{2}-1}}$	$\|x\|<1$

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