transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Several numbers are named after Euler.
Euler’s number is the irrational number typically denoted “e” and defined by the series
This happens to also be the groupoid cardinality of the groupoid $core(FinSet)$, the core of FinSet.
The Euler numbers (also called secant numbers) $E_n$, $n\geq 0$ are a sequence of integers defined via the generating function
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.
Sometimes Euler numbers are defined as a different sequence that includes tangent numbers (and is non-alternating in sign):
where one has $1, 1, 1, 2, 5, 16, \ldots$ as the first few such Euler numbers. These $E_n$ satisfy the recurrence
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
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
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
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
For applications in algebraic geometry, see “Eulerian polynomials-[Hirzebruch]”.
There is also a number, usually denoted as $\gamma$, called the Euler-Mascheroni constant. It is defined as a limit
where $\log$ 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.