Euler numbers

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In mathematics, the Euler numbers are a sequence En of integers (sequence A122045 in the OEIS) defined by the Taylor series expansion

,

where cosh t is the hyperbolic cosine. The Euler numbers are related to a special value of the Euler polynomials, namely:

The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. They also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.

Examples[edit]

The odd-indexed Euler numbers are all zero. The even-indexed ones (sequence A028296 in the OEIS) have alternating signs. Some values are:

E0 = 1
E2 = −1
E4 = 5
E6 = −61
E8 = 1385
E10 = −50521
E12 = 2702765
E14 = −199360981
E16 = 19391512145
E18 = −2404879675441

Some authors re-index the sequence in order to omit the odd-numbered Euler numbers with value zero, or change all signs to positive (sequence A000364 in the OEIS). This article adheres to the convention adopted above.

Explicit formulas[edit]

In terms of Stirling numbers of the second kind[edit]

Following two formulas express the Euler numbers in terms of Stirling numbers of the second kind[1] [2]

where denotes the Stirling numbers of the second kind, and denotes the rising factorial.

As a double sum[edit]

Following two formulas express the Euler numbers as double sums[3]

As an iterated sum[edit]

An explicit formula for Euler numbers is:[4]

where i denotes the imaginary unit with i2 = −1.

As a sum over partitions[edit]

The Euler number E2n can be expressed as a sum over the even partitions of 2n,[5]

as well as a sum over the odd partitions of 2n − 1,[6]

where in both cases K = k1 + ··· + kn and

is a multinomial coefficient. The Kronecker deltas in the above formulas restrict the sums over the ks to 2k1 + 4k2 + ··· + 2nkn = 2n and to k1 + 3k2 + ··· + (2n − 1)kn = 2n − 1, respectively.

As an example,

As a determinant[edit]

E2n is given by the determinant

As an integral[edit]

E2n is also given by the following integrals:

Congruences[edit]

W. Zhang[7] obtained the following combinational identities concerning the Euler numbers, for any prime , we have

W. Zhang and Z. Xu[8] proved that, for any prime and integer , we have

where is the Euler's totient function.

Asymptotic approximation[edit]

The Euler numbers grow quite rapidly for large indices as they have the following lower bound

Euler zigzag numbers[edit]

The Taylor series of is

where An is the Euler zigzag numbers, beginning with

1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832, ... (sequence A000111 in the OEIS)

For all even n,

where En is the Euler number; and for all odd n,

where Bn is the Bernoulli number.

For every n,

[citation needed]

See also[edit]

References[edit]

  1. ^ Jha, Sumit Kumar (2019). "A new explicit formula for Bernoulli numbers involving the Euler number". Moscow Journal of Combinatorics and Number Theory. 8 (4): 385–387. doi:10.2140/moscow.2019.8.389.
  2. ^ Jha, Sumit Kumar (15 November 2019). "A new explicit formula for the Euler numbers in terms of the Stirling numbers of the second kind".
  3. ^ Wei, Chun-Fu; Qi, Feng (2015). "Several closed expressions for the Euler numbers". Journal of Inequalities and Applications. 219 (2015). doi:10.1186/s13660-015-0738-9.
  4. ^ Tang, Ross (2012-05-11). "An Explicit Formula for the Euler zigzag numbers (Up/down numbers) from power series" (PDF).
  5. ^ Vella, David C. (2008). "Explicit Formulas for Bernoulli and Euler Numbers". Integers. 8 (1): A1.
  6. ^ Malenfant, J. (2011). "Finite, Closed-form Expressions for the Partition Function and for Euler, Bernoulli, and Stirling Numbers". arXiv:1103.1585 [math.NT].
  7. ^ Zhang, W.P. (1998). "Some identities involving the Euler and the central factorial numbers" (PDF). Fibonacci Quarterly. 36 (4): 154–157.
  8. ^ Zhang, W.P.; Xu, Z.F. (2007). "On a conjecture of the Euler numbers". Journal of Number Theory. 127 (2): 283–291. doi:10.1016/j.jnt.2007.04.004.

External links[edit]