Difference between revisions of "Genocchi numbers"
From specialfunctionswiki
(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...") |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 3: | Line 3: | ||
=Properties= | =Properties= | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
<div class="toccolours mw-collapsible mw-collapsed"> | <div class="toccolours mw-collapsible mw-collapsed"> | ||
<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> | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 18:57, 24 May 2016
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 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: █
