nLab Euler number

Several numbers are named after Euler.

Euler’s number

Euler’s number is the irrational number typically denoted “e” and defined by the series

$$e \coloneqq \sum_{n \in \mathbb{N}} \frac{1}{n!} \,.$$

This happens to also be the groupoid cardinality of the groupoid $core(FinSet)$, the core of FinSet.

Euler numbers

The Euler numbers (also called secant numbers) $E_n$, $n\geq 0$ are a sequence of integers defined via the generating function

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

All odd-numbered members of the sequence vanish: $E_{2k-1}=0$ for $k\in\mathbb{N}$. $E_0=1$, $E_2 = -1$, $E_4 = 5$, see Euler number at wikipedia for more and the Abramowitz–Stegun handbook for many Euler numbers.

Remark

Sometimes Euler numbers are defined as a different sequence that includes tangent numbers (and is non-alternating in sign):

$$\sec(x) + \tan(x) = \sum_{n \geq 0} E_n \frac{x^n}{n!}$$

where one has $1, 1, 1, 2, 5, 16, \ldots$ as the first few such Euler numbers. These $E_n$ satisfy the recurrence

$$2 E_{n+1} = \sum_i \binom{n}{i} E_i E_{n-i}$$

and count the number of alternating permutations $\pi$ on $\{1, 2, \ldots, n\}$, i.e. permutations such that $\pi(1) \gt \pi(2) \lt \pi(3) \gt \ldots$. See for example this $n$-Category Café post.

Euler numbers generalize to Euler polynomials $E_n(x)$ defined via the generating function

$$\frac{2e^{tx}}{e^t+1}=\sum_{n\geq 0} E_n(x) \frac{t^n}{n!},$$

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

$$E_n(x) = \sum_k \left(\array{n\\ k}\right) \frac{E_k}{2^k}\left(x-\frac{1}{2}\right)^{n-k}$$

There is also a complementarity formula $E_n(1-x)=(-1)^n E_n(x)$. 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 $A(n,k)$). Combinatorially, $A(n, k)$ counts the number of permutations $\pi$ of the set $\{1, 2, \ldots, n\}$ with $k$ ascents (an ascent of $\pi$ is an element $j$ such that $\pi(j) \lt \pi(j+1)$). Alternatively, the double sequence can be defined recursively through the formula

$$x^n = \sum_k A(n, k) \binom{x+k}{n}.$$

For applications in algebraic geometry, see “Eulerian polynomials-[Hirzebruch]”.

Euler-Mascheroni constant

There is also a number, usually denoted as $\gamma$, called the Euler-Mascheroni constant. It is defined as a limit

$$\gamma \coloneqq \lim_{n \to \infty} \sum_{k=1}^n \frac1{k} - \log n$$

where $\log$ is the natural logarithm. It arises for example in discussions of the Gamma function.

Related concepts

• exponential map

• e

• pi

• golden ratio