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 n1n!. e \coloneqq \sum_{n \in \mathbb{N}} \frac{1}{n!} \,.

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

Euler numbers

The Euler numbers (also called secant numbers) E nE_n, n0n\geq 0 are a sequence of integers defined via the generating function

1cht=2e t+e t= n0E nt nn! \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 2k1=0E_{2k-1}=0 for kk\in\mathbb{N}. E 0=1E_0=1, E 2=1E_2 = -1, E 4=5E_4 = 5, see Euler number at wikipedia for more and the Abramowitz–Stegun handbook for many Euler numbers.


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

sec(x)+tan(x)= n0E nx nn!\sec(x) + \tan(x) = \sum_{n \geq 0} E_n \frac{x^n}{n!}

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

2E n+1= i(ni)E iE ni2 E_{n+1} = \sum_i \binom{n}{i} E_i E_{n-i}

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

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

2e txe t+1= n0E n(x)t nn!, \frac{2e^{tx}}{e^t+1}=\sum_{n\geq 0} E_n(x) \frac{t^n}{n!},

so that E n=2 nE n(12)E_n = 2^n E_n(\frac{1}{2}). The Euler polynomials satisfy the recursion E n(x+1)+E n(x)=2x nE_n(x+1)+E_n(x)=2x^n and may be conversely expressed via Euler numbers as

E n(x)= k(n k)E k2 k(x12) nk 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(1x)=(1) nE n(x)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)A(n,k)). Combinatorially, A(n,k)A(n, k) counts the number of permutations π\pi of the set {1,2,,n}\{1, 2, \ldots, n\} with kk ascents (an ascent of π\pi is an element jj such that π(j)<π(j+1)\pi(j) \lt \pi(j+1)). Alternatively, the double sequence can be defined recursively through the formula

x n= kA(n,k)(x+kn).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 γ\gamma, called the Euler-Mascheroni constant. It is defined as a limit

γlim n k=1 n1klogn\gamma \coloneqq \lim_{n \to \infty} \sum_{k=1}^n \frac1{k} - \log n

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

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