nLab Euler number

Several numbers are named after Euler.

Contents

Euler’s number

Euler’s number is the irrational number

$e:=\sum _{n\in ℕ}\frac{1}{n!}\phantom{\rule{thinmathspace}{0ex}}.$e := \sum_{n \in \mathbb{N}} \frac{1}{n!} \,.

This is the groupoid cardinality of the groupoid $\mathrm{core}\left(\mathrm{FinSet}\right)$, the core of FinSet.

Euler numbers

The Euler numbers ${E}_{n}$, $n\ge 0$ are a sequence of integers defined via the generating function

$\frac{1}{\mathrm{ch}t}=\frac{2}{{e}^{t}+{e}^{-t}}=\sum _{n\ge 0}{E}_{n}\frac{{t}^{n}}{n!}$\frac{1}{ch t}= \frac{2}{e^t+e^{-t}} = \sum_{n\geq 0} E_n \frac{t^n}{n!}

All odd-numbered members of the sequence vanish: ${E}_{2k-1}=0$ for $k\in ℕ$. ${E}_{0}=1$, ${E}_{2}=-1$, ${E}_{4}=5$, see Euler number at wikipedia for more and the Abramowitz–Stegun handbook for many Euler numbers.

Euler numbers generalize to Euler polynomials ${E}_{n}\left(x\right)$ defined via the generating function

$\frac{2{e}^{\mathrm{tx}}}{{e}^{t}+1}=\sum _{n\ge 0}{E}_{n}\left(x\right)\frac{{t}^{n}}{n!},$\frac{2e^{tx}}{e^t+1}=\sum_{n\geq 0} E_n(x) \frac{t^n}{n!},

so that ${E}_{n}={2}^{n}{E}_{n}\left(\frac{1}{2}\right)$. The Euler polynomials satisfy the recursion ${E}_{n}\left(x+1\right)+{E}_{n}\left(x\right)=2{x}^{n}$ and may be conversely expressed via Euler numbers as

${E}_{n}\left(x\right)=\sum _{k}\left(\begin{array}{c}n\\ k\end{array}\right)\frac{{E}_{k}}{{2}^{k}}{\left(x-\frac{1}{2}\right)}^{n-k}$E_n(x) = \sum_k \left(\array{n\\ k}\right) \frac{E_k}{2^k}\left(x-\frac{1}{2}\right)^{n-k}

There is also a complementarity formula ${E}_{n}\left(1-x\right)=\left(-1{\right)}^{n}{E}_{n}\left(x\right)$. See e.g. Louis Comtet, Advanced combinatorics, D. Reifel Publ. Co., Dordrecht-Holland, Boston 1974 djvu

Eulerian numbers

There are also Eulerian numbers (forming a different, double sequence $A\left(n,k\right)$).

Revised on May 13, 2010 12:34:55 by Urs Schreiber (87.212.203.135)