Recurrence Relation for Euler Numbers
(Redirected from Sum of Euler Numbers by Binomial Coefficients Vanishes/Corollary)
Jump to navigation
Jump to search
Theorem
Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Then:
| \(\ds E_{2 n}\) | \(=\) | \(\ds -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\binom {2 n} 0 E_0 + \binom {2 n} 2 E_2 + \binom {2 n} 4 E_4 + \cdots + \binom {2 n} {2 n - 2} E_{2 n - 2} }\) |
where $E_n$ denotes the $n$th Euler number.
Proof
| \(\ds \forall n \in \Z_{>0}: \, \) | \(\ds \sum_{k \mathop = 0}^n \binom {2 n} {2 k} E_{2 k}\) | \(=\) | \(\ds 0\) | Sum of Euler Numbers by Binomial Coefficients Vanishes | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k} + \dbinom {2 n} {2 n} E_{2 n}\) | \(=\) | \(\ds 0\) | separating out case where $k = n$ | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds E_{2 n}\) | \(=\) | \(\ds -\sum_{k \mathop = 0}^{n - 1} \dbinom {2 n} {2 k} E_{2 k}\) | Binomial Coefficient with Self: $\dbinom {2 n} {2 n} = 1$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 21$: Relationships of Bernoulli and Euler Numbers: $21.6$