Difference between revisions of "Genocchi numbers"
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(Created page with "The Genocchi numbers $G_n$ are given by the generating function $$\dfrac{2t}{e^t+1} = \displaystyle\sum_{k=0}^{\infty} G_n \dfrac{t^n}{n!}.$$ =Properties= <div class="toccolo...") |
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<strong>Proposition:</strong> The following formula holds: | <strong>Proposition:</strong> The following formula holds: | ||
$$G_{2n}=2(1-2^{2n})B_{2n}= 2nE_{2n-1}(0),$$ | $$G_{2n}=2(1-2^{2n})B_{2n}= 2nE_{2n-1}(0),$$ | ||
| − | where $G_{2n}$ denotes [[Genocchi numbers]], $B_{2n}$ denotes [[Bernoulli numbers]], and $E_{2n-1}$ denotes [[Euler | + | where $G_{2n}$ denotes [[Genocchi numbers]], $B_{2n}$ denotes [[Bernoulli numbers]], and $E_{2n-1}$ denotes an [[Euler polynomial]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 20:03, 23 March 2015
The Genocchi numbers $G_n$ are given by the generating function $$\dfrac{2t}{e^t+1} = \displaystyle\sum_{k=0}^{\infty} G_n \dfrac{t^n}{n!}.$$
Properties
Proposition: The following values hold for the Genocchi numbers: $$G_1=1, G_3=G_5=G_7=G_9=G_11=\ldots=0.$$
Proof: █
Proposition: The following formula holds: $$G_{2n}=2(1-2^{2n})B_{2n}= 2nE_{2n-1}(0),$$ where $G_{2n}$ denotes Genocchi numbers, $B_{2n}$ denotes Bernoulli numbers, and $E_{2n-1}$ denotes an Euler polynomial.
Proof: █
