nLab Euler number

Redirected from "Euler numbers".

Contents

Context

Arithmetic

Several numbers are named after Euler.

Euler’s number
Euler numbers
Eulerian numbers
Euler-Mascheroni constant
Related concepts

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

Last revised on November 21, 2023 at 01:22:06. See the history of this page for a list of all contributions to it.