Euler numbers
The Euler numbers E_n are a sequence of integers in mathematics, defined such that E_n = 0 for all odd n \geq 1, with the even-indexed terms alternating in sign and growing rapidly: E_0 = 1, E_2 = -1, E_4 = 5, E_6 = -61, E_8 = 1385, E_{10} = -50521, and so on. They arise as the coefficients in the Taylor series expansion of the hyperbolic secant function, given by the exponential generating function \sech x = \sum_{n=0}^\infty E_n \frac{x^n}{n!}, where \sech x = \frac{2}{e^x + e^{-x}} for |x| < \pi/2.[1] Named after the Swiss mathematician Leonhard Euler, who first explored their properties through series expansions of trigonometric and hyperbolic functions in the mid-18th century, the Euler numbers have since been studied extensively in analysis, number theory, and combinatorics. They satisfy various recurrence relations and have explicit formulas involving Stirling numbers of the second kind.[2] Euler numbers exhibit deep connections to other mathematical objects, including Bernoulli numbers, and appear in the evaluation of the Riemann zeta function at even integers through \zeta(2n) = (-1)^{n+1} \frac{(2\pi)^{2n} B_{2n}}{2 (2n)!}, linking to Euler's original work on infinite series. In combinatorics, the absolute values |E_{2n}| enumerate the number of alternating permutations of $2n+1 elements, while asymptotic approximations describe their growth as E_{2n} \sim (-1)^n \frac{4^{n+1} (2n)!}{\pi^{2n+1}}. These properties make Euler numbers fundamental in enumerative combinatorics, special function theory, and modular form congruences.[2][3]Definition and Basics
Definition
The Euler numbers E_n form an integer sequence defined as the coefficients in the Taylor series expansion of the hyperbolic secant function around t = 0: \sech t = \sum_{n=0}^\infty \frac{E_n}{n!} t^n, \quad |t| < \frac{\pi}{2}. [4] This expansion holds with E_n = 0 for all odd n > 0, so only the even-indexed terms contribute nonzero values.[4] The first few nonzero Euler numbers are E_0 = 1, E_2 = -1, E_4 = 5, and E_6 = -61, illustrating the alternating signs for even indices where the sign of E_{2k} is (-1)^k.[4] In standard notation, E_n denotes this signed integer sequence, distinct from unsigned variants (often denoted |E_n| or zigzag numbers) that appear in related expansions such as the secant and tangent functions.[5] Combinatorially, the Euler numbers provide a signed enumeration related to the count of alternating permutations of sets.[6] The Euler numbers relate to the Euler polynomials E_n(x) via the evaluation E_n = 2^n E_n\left(\frac{1}{2}\right).[4]Historical Background
Leonhard Euler first encountered the coefficients now known as Euler numbers during his investigations into the infinite series expansions of trigonometric functions in the mid-18th century. In his seminal work Introductio in analysin infinitorum published in 1748, Euler derived power series for functions such as secant and tangent, where these integer coefficients emerged naturally from the reciprocal of the cosine series and related manipulations. These appearances were motivated by Euler's broader efforts to systematize infinite series and their applications in analysis, building on earlier work with Bernoulli numbers for similar expansions of cotangent and other functions. The coefficients remained unnamed in Euler's original presentations, appearing simply as constants in the series for sec(x) and tan(x). It was not until the 19th century that they received formal recognition and nomenclature. Heinrich Friedrich Scherk first referred to them as "Euler's numbers" in 1825, specifically in connection with the secant series coefficients.[7] This terminology was later adopted and expanded by mathematicians such as Joseph Ludwig Raabe in 1851, who applied it to multiples of the secant numbers, and James Whitbread Lee Glaisher, who in works like his 1878 paper on expressions for Bernoulli and Eulerian numbers provided determinant representations and linked them to finite differences and combinatorial contexts.[8] It is important to distinguish these Euler numbers—focused here on the secant and tangent series coefficients—from other mathematical objects bearing Euler's name, such as the totient function φ(n), which Euler introduced in the 1760s to count integers coprime to n in number theory.[9] Similarly, the Euler characteristic in topology, defined later in the 19th century, represents a different invariant. The notation has evolved from Euler's ad hoc constants to the modern E_n, where E_{2k+1} = 0 for odd indices and even-indexed terms alternate in sign (e.g., E_0 = 1, E_2 = -1, E_4 = 5), a convention standardized in the late 19th and early 20th centuries to align with the Taylor series of sech(x).[2] This shift addressed earlier variations in sign placements across different series expansions.Properties
Basic Properties
The Euler numbers E_n vanish for all odd indices n \geq 1, so E_n = 0 whenever n is odd.[10] Thus, the non-zero Euler numbers occur exclusively at even indices.[11] The even-indexed Euler numbers exhibit an alternating sign pattern, beginning with E_0 = 1 > 0, followed by E_2 = -1 < 0, E_4 = 5 > 0, E_6 = -61 < 0, and continuing in this manner.[11] All Euler numbers are integers.[11] The absolute values |E_{2k}| of the even-indexed terms grow factorially with k.[12] A fundamental identity is E_0 = 1.[11] Moreover, the Euler numbers coincide with the Euler polynomials evaluated at zero: E_n = E_n(0) for all n \geq 0.[11] For intuition, the even-powered terms relate to the Taylor series of the hyperbolic secant function \operatorname{sech}(t).[12] The Euler numbers are odd integers, meaning they are not divisible by 2.[11]Congruences and Divisibility
The modular arithmetic of Euler numbers reveals significant patterns, particularly modulo primes. For an odd prime p, the Euler number E_{p-1} satisfies E_{p-1} \equiv 0 \pmod{p} if p \equiv 1 \pmod{4}, and E_{p-1} \equiv 2 \pmod{p} if p \equiv 3 \pmod{4}.[13] This follows from the general divisibility rule: if (p-1) \mid 2n, then E_{2n} \equiv 0 \pmod{p} when p \equiv 1 \pmod{4}, and E_{2n} \equiv 2 \pmod{p} when p \equiv 3 \pmod{4}.[13] For example, with p=7 \equiv 3 \pmod{4} and $2n = p-1 = 6, E_6 = -61 \equiv 2 \pmod{7}. Similarly, for p=5 \equiv 1 \pmod{4}, E_4 = 5 \equiv 0 \pmod{5}. Kummer congruences provide a framework for relating Euler numbers across arithmetic progressions modulo primes. For an odd prime p and integer n \geq 2, E_n \equiv E_{n + p - 1} \pmod{p}.[13] More generally, Carlitz established Kummer-type congruences for Euler numbers, such as \sum_{s=0}^{r-1} (-1)^s \binom{r}{s} E_{n + s(p-1)} \equiv 0 \pmod{p^r} for r > 1, n > r, and odd prime p.[14] These extend classical results and facilitate computations modulo prime powers. For instance, E_4 \equiv 5 \pmod{7}, which aligns with direct evaluation but can be verified via such relations for higher indices. Divisibility properties of Euler numbers by primes are tied to specific conditions. Primes p \equiv 1 \pmod{4} always divide E_{p-1}, as noted above. For higher powers, p^\ell \mid E_{2n} under the condition (p-1)p^{\ell-1} \mid 2n when p \equiv 1 \pmod{4}, with analogous rules modulo p^\ell for p \equiv 3 \pmod{4}.[13] Euler numbers connect to irregular primes through an analogous relation to Bernoulli numbers. Irregular primes p are those dividing the numerator of some B_k (with $2 \leq k \leq p-3) after clearing denominators, per Kummer's criterion for Fermat's Last Theorem. Primes dividing certain E_m (e.g., E_{p-3}, E_{p-5}) are termed irregular relative to Euler numbers, with Vandiver showing implications for the first case of Fermat's Last Theorem if no such divisibility holds.[14] Computations confirm infinitely many irregular primes divide Euler numbers like E_2, E_4, \dots, E_{p-3}.[14]Computation and Examples
Numerical Examples
The Euler numbers E_n for small indices provide concrete illustrations of their values, with E_n = 0 for all odd n \geq 1. The following table lists the values from n=0 to n=20:| n | E_n |
|---|---|
| 0 | 1 |
| 1 | 0 |
| 2 | -1 |
| 3 | 0 |
| 4 | 5 |
| 5 | 0 |
| 6 | -61 |
| 7 | 0 |
| 8 | 1385 |
| 9 | 0 |
| 10 | -50521 |
| 11 | 0 |
| 12 | 2702765 |
| 13 | 0 |
| 14 | -199360981 |
| 15 | 0 |
| 16 | 19391512145 |
| 17 | 0 |
| 18 | -2404879675441 |
| 19 | 0 |
| 20 | 370371188237525 |
Asymptotic Bounds
The asymptotic behavior of the Euler numbers E_{2n} for large n is characterized by super-exponential growth dominated by a factorial term modulated by powers of \pi. A fundamental asymptotic formula is (-1)^n E_{2n} \sim \frac{2^{2n+2} (2n)!}{\pi^{2n+1}} as n \to \infty. This expression, derived from the singularity analysis of the generating function \sec x, captures the leading-order magnitude |E_{2n}| \approx \frac{4 \cdot 4^n (2n)!}{\pi^{2n+1}}.[15] Equivalently, incorporating Stirling's approximation for (2n)! yields a refined form emphasizing the explicit exponential and polynomial factors: (-1)^n E_{2n} \sim 8 \sqrt{\frac{n}{\pi}} \left( \frac{4n}{\pi e} \right)^{2n}. This approximation improves accuracy for moderate n by accounting for the subdominant \sqrt{n} prefactor, with the base \frac{4n}{\pi e} exceeding 1 for n \gtrsim [3](/page/3), driving the overall growth. The relative error in this expansion decreases as n increases, providing both lower and upper bounds; for instance, for sufficiently large n, |E_{2n}| > \frac{8}{\pi} \sqrt{\frac{n}{\pi}} \left( \frac{4n}{\pi e} \right)^{2n}, since the leading coefficient 8 exceeds \frac{8}{\pi} \approx 2.546 and the ratio to the full asymptotic approaches 1.[15][2] In relation to factorial growth, the normalized sequence satisfies \frac{|E_{2n}|}{(2n)!} \sim \frac{4}{\pi} \left( \frac{2}{\pi} \right)^{2n}, indicating that |E_{2n}| grows like (2n)! scaled by a factor that decays double-exponentially due to \left( \frac{2}{\pi} \right)^{2n} < 1. This limit arises directly from substituting Stirling's formula into the first asymptotic expression, confirming the constant \frac{4}{\pi} \approx 1.273 as the precise scaling.[15] Post-2000 refinements enhance precision for computational purposes. A notable improvement adjusts the base term for higher accuracy: (-1)^n E_{2n} \sim 8 \sqrt{\frac{n}{\pi}} \left( \frac{4n}{\pi e} \cdot \frac{480n^2 + 9}{480n^2 - 1} \right)^{2n}, which incorporates quadratic corrections in n and yields significantly more decimal digits than the basic form for large even indices (e.g., over 18 digits for n=500). This expansion facilitates tighter inclusions, such as |E_{1000}| \approx 0.3887561841253070615 \times 10^{2372}, bounding the value between consecutive powers of 10 with minimal error.[2]Generating Functions
Secant and Tangent Series
The hyperbolic secant function provides an exponential generating function for the Euler numbers, which vanish for odd indices greater than zero, emphasizing its even-powered expansion. The Taylor series is given by \sech t = \sum_{n=0}^{\infty} E_n \frac{t^n}{n!}, where E_0 = 1, E_2 = -1, E_4 = 5, E_6 = -61, and subsequent even-indexed terms alternate in sign with increasing magnitude. This series converges for |t| < \pi/2.[2] An equivalent form arises from the secant function via the identity \sec t = \sech(it), leading to the expansion \sec t = \sum_{k=0}^{\infty} (-1)^k E_{2k} \frac{t^{2k}}{(2k)!}, which simplifies to \sec t = \sum_{k=0}^{\infty} |E_{2k}| \frac{t^{2k}}{(2k)!} using the signed convention for E_{2k}, with coefficients yielding positive terms such as $1 + \frac{1}{2} t^2 + \frac{5}{24} t^4 + \frac{61}{720} t^6 + \cdots. This converges for |t| < \pi/2. The secant numbers |E_{2k}| (also denoted S_k) are thus the absolute values of the even Euler numbers.[2][16] The combination of secant and tangent functions generates a broader series incorporating both even and odd indices through related Euler numbers E_n^*, defined by \tan t + \sec t = \sum_{n=0}^{\infty} E_n^* \frac{t^n}{n!}, where E_0^* = 1, E_1^* = 1, E_2^* = 1, E_3^* = 2, E_4^* = 5, E_5^* = 16, E_6^* = 61, and these values count alternating permutations, with even indices matching the secant coefficients and odd indices the tangent numbers. These E_n^* are variants of the standard even Euler numbers, extended to nonzero odd terms, and the series converges for |t| < \pi/2.[17][16] These generating functions can be derived by power series manipulation of \cosh t = \sum_{k=0}^{\infty} \frac{t^{2k}}{(2k)!} to obtain \sech t via inversion, or by solving the differential equation y' = \sec t \cdot y with initial condition y(0) = 1 for y = \tan t + \sec t. The former involves recursive division of formal power series, while the latter uses the known derivatives of trigonometric functions to equate coefficients.[2][17]Exponential Generating Function
The exponential generating function for the Euler numbers E_n is given by \sech t = \sum_{n=0}^{\infty} E_n \frac{t^n}{n!}. This representation underscores the even symmetry of the sequence, as \sech t is an even function, implying E_n = 0 for all odd n \geq 1. The egf encapsulates key analytic properties, such as the rapid growth of |E_n| reflected in the poles of \sech t near the imaginary axis, and supports derivations of recurrence relations through differentiation of the generating function.[18] An alternative exponential generating function connects the Euler numbers to the Euler zigzag numbers \tilde{E}_n via analytic continuation. Specifically, the egf for the zigzag numbers is \sec t + \tan t = \sum_{n=0}^{\infty} \tilde{E}_n \frac{t^n}{n!}, and substituting t \to it yields \sec(it) + \tan(it) = \sech t + i \tanh t, where the real part aligns with the Euler egf; variants like (\sec t + \tan t) e^{it} appear in complex extensions for signed counts but preserve the core structure for even indices.[18][19] In combinatorics, the egf \sec t + \tan t aligns with the exponential formula, enumerating structures on labeled sets. The coefficient \tilde{E}_n equals the number of alternating (up-down) permutations of $$, providing a bijective link to the values; this extends to counting signed permutations with fixed ascent patterns or binary search trees with alternating labels via exponential composition. For the classical Euler numbers, the connection via it-substitution translates these to even-length structures, such as complete matchings in signed graphs.[19][20]Explicit Formulas
Recursive Formulas
The Euler numbers E_n satisfy the recurrence relation \sum_{k=0}^n \binom{n}{k} 2^k E_{n-k} + E_n = 2 for n \geq 0, with the initial condition E_0 = 1.[21] This equation can be rearranged to solve explicitly for E_n: E_n = 1 - \frac{1}{2} \sum_{k=1}^n \binom{n}{k} 2^k E_{n-k}. The relation originates from properties of the generating function \sum_{n=0}^\infty E_n \frac{x^n}{n!} = \mathrm{sech}(x).[4] Since E_n = 0 for all odd n \geq 1, the formula simplifies for even indices. For n = 2m with m \geq 1, \sum_{\substack{k=0 \\ k \text{ even}}}^{2m} \binom{2m}{k} 2^k E_{2m-k} + E_{2m} = 2, where the sum is over even k because terms with odd k vanish due to E_{2m-k} = 0 when $2m - k is odd. This reduced form involves only previous even-indexed Euler numbers, making it efficient for computing the sequence of nonzero terms.[21] These recurrences enable computation of E_n via dynamic programming, where each E_n is calculated from prior values in O(n) time, yielding an overall complexity of O(n^2) to compute up to E_n. For verification, applying the formula for small n:- For n=1: E_1 + 2 E_0 + E_1 = 2 simplifies to $2E_1 + 2 = 2, so E_1 = 0.
- For n=2: E_2 + 4 E_0 + E_2 = 2 simplifies to $2E_2 + 4 = 2, so E_2 = -1.
- For n=4: E_4 + 16 E_0 - 24 E_2 + E_4 = 2 simplifies to $2E_4 - 8 = 2, so E_4 = 5.[21]