Open Access

Several closed expressions for the Euler numbers

Journal of Inequalities and Applications20152015:219

https://doi.org/10.1186/s13660-015-0738-9

Received: 11 November 2014

Accepted: 17 June 2015

Published: 2 July 2015

Abstract

In the paper, the authors establish several closed expressions for the Euler numbers in the form of a determinant or double sums and in terms of, for example, the Stirling numbers of the second kind.

Keywords

Euler numberclosed expressiondeterminantStirling number of the second kinddouble sum

MSC

11B6811B7333B10

1 Introduction and main results

It is well known ([1], p.75, item 4.3.69) that the secant function secz may be expanded at \(z=0\) into the power series
$$ \sec z=\sum_{k=0}^{\infty}(-1)^{k}E_{2k} \frac{z^{2k}}{(2k)!},\quad |z|< \frac{\pi}{2}, $$
(1.1)
where \(E_{k}\) are called in number theory the Euler numbers which may also be defined ([2], p.15) by
$$ \frac{1}{\cosh z}=\frac{2e^{z}}{e^{2z}+1} =\sum _{k=0}^{\infty}E_{k}\frac{z^{k}}{k!} =\sum _{k=0}^{\infty}E_{2k} \frac{z^{2k}}{(2k)!},\quad |z|< \frac{\pi}{2}. $$
(1.2)
In number theory, the numbers
$$ S_{k}=(-1)^{k}E_{2k} $$
(1.3)
are called in [3], p.128, for example, the secant numbers or the zig numbers.

These numbers also occur in combinatorics, specifically when counting the number of alternating permutations of a set with an even number of elements.

The first few secant numbers \(S_{k}\) for \(k=1,2,3,4\) are 1, 5, 61, \(1\text{,}385\). The first few Euler numbers \(E_{2k}\) for \(0\le k\le9\) are
$$\begin{aligned} \begin{aligned} &E_{0} = 1,\qquad E_{2} = -1,\qquad E_{4} = 5, \qquad E_{6} = -61,\qquad E_{8} = 1\text{,}385, \\ &E_{10} = -50\text{,}521,\qquad E_{12} = 2\text{,}702 \text{,}765,\qquad E_{14} = -199\text{,}360\text{,}981, \\ &E_{16} = 19\text{,}391\text{,}512\text{,}145,\qquad E_{18} = -2\text{,}404\text{,}879\text{,}675\text{,}441. \end{aligned} \end{aligned}$$

It is a classical topic to find closed expressions for the Euler numbers \(E_{2k}\) and the tangent number \(S_{k}\). These numbers are closely connected with many other numbers and functions, such as the Bernoulli numbers, the Genocchi numbers, the tangent numbers, the Euler polynomials, the Stirling numbers of two kinds, and the Riemann zeta function, in number theory and combinatorics. There has been a plenty of literature such as [1, 2, 410] and closely related references therein.

In mathematics, a closed expression is a mathematical expression that can be evaluated in a finite number of operations. It may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit.

In this paper, we establish several closed expressions for the Euler numbers \(E_{2k}\) in the form of a determinant of order 2k or double sums and in terms of, for example, the Stirling numbers of the second kind \(S(n,k)\) which may be generated ([2], p.20) by
$$ \frac{(e^{x}-1)^{k}}{k!}=\sum_{n=k}^{\infty}S(n,k) \frac{x^{n}}{n!},\quad k\in\mathbb{N}, $$
may be computed ([2], p.21) by the closed expression
$$ S(k,m)=\frac{1}{m!}\sum_{\ell=1}^{m}(-1)^{m-\ell} \binom{m}{\ell}\ell ^{k}, \quad 1\le m\le k, $$
and may be interpreted combinatorially as the number of ways of partitioning a set of n elements into k nonempty subsets.

Our main results may be formulated as the following theorems.

Theorem 1.1

For \(k\in\mathbb{N}\),
$$ E_{2k}=(-1)^{k} \biggl| \binom{i}{j-1}\cos \biggl((i-j+1)\frac{\pi}{2} \biggr) \biggr| _{(2k)\times(2k)}, $$
where \(|c_{ij}|_{k\times k}\) is the determinant of a matrix \([c_{ij}]_{k\times k}\) of elements \(c_{ij}\) and order k.

Theorem 1.2

For \(k\in\mathbb{N}\),
$$ E_{2k}=(2 k+1)\sum_{\ell=1}^{2k} (-1)^{\ell}\frac{1}{2^{\ell}(\ell +1)}\binom{2 k}{\ell}\sum _{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2k}. $$
(1.4)

Theorem 1.3

For \(n\in\mathbb{N}\),
$$ E_{n}=1+\sum_{k=1}^{n} \frac{(k+1)!}{2^{k}}S(n,k)\sum_{\ell=1}^{k}(-1)^{\ell } \frac{2^{\ell}}{\ell+1} \binom{\ell+1}{k-\ell} $$
(1.5)
and
$$ E_{n}=1+\sum_{\ell=1}^{n}(-1)^{\ell} \frac{1}{\ell+1}\sum_{k=0}^{n-\ell } \frac{(k+\ell+1)!}{2^{k}} \binom{\ell+1}{k}S(n,k+\ell). $$
(1.6)

Theorem 1.4

For \(k\in\mathbb{N}\),
$$ E_{2k}=\sum_{i=1}^{2k}(-1)^{i} \frac{1}{2^{i}}\sum_{\ell=0}^{2i}(-1)^{\ell } \binom{2i}{\ell}(i-\ell)^{2k}. $$
(1.7)

2 Lemmas

In order to prove our main results, we need the following lemmas.

Lemma 2.1

Let \(u=u(x)\) and \(v=v(x)\ne0\) be differentiable functions, let \(U_{n+1,1}\) be an \((n+1)\times1\) matrix whose elements \(u_{k,1}=u^{(k-1)}(x)\) for \(1\le k\le n+1\), let \(V_{n+1,n}\) be an \((n+1)\times n\) matrix whose elements \(v_{i,j}=\binom {i-1}{j-1}v^{(i-j)}(x)\) for \(1\le i\le n+1\) and \(1\le j\le n\), and let \(|W_{n+1,n+1}|\) denote the determinant of the \((n+1)\times(n+1)\) matrix \(W_{n+1,n+1}=[U_{n+1,1} \ V_{n+1,n}]\). Then the nth derivative of the ratio \(\frac{u(x)}{v(x)}\) may be computed by
$$ \frac{\mathrm {d}^{n}}{\mathrm {d}x^{n}} \biggl(\frac{u}{v} \biggr)=(-1)^{n} \frac {|W_{n+1,n+1}|}{v^{n+1}}. $$

Proof

This is a reformulation of [11], p.40, Exercise 5. □

The Bell polynomials of the second kind \(\mathrm{B}_{n,k}\), or say, the partial Bell polynomials \(\mathrm{B}_{n,k}\), may be defined ([12], p.134, Theorem A) by
$$ \mathrm{B}_{n,k}(x_{1},x_{2},\ldots,x_{n-k+1})= \sum_{\substack{1\le q\le n,\ell_{q}\in\{0\}\cup\mathbb{N}\\ \sum_{q=1}^{n}i\ell_{q}=n\\ \sum_{q=1}^{n}\ell_{q}=k}}\frac{n!}{\prod_{q=1}^{n-k+1}\ell_{q}!} \prod _{q=1}^{n-k+1} \biggl(\frac{x_{q}}{q!} \biggr)^{\ell_{q}} $$
for \(n\ge k\ge0\).

Lemma 2.2

(Faà di Bruno formula [12], p.139, Theorem C)

For \(n\in\mathbb{N}\), the nth derivative of a composite function \(f(g(x))\) may be computed in terms of the Bell polynomials of the second kind \(\mathrm{B}_{n,k}\) by
$$ \frac{\mathrm {d}^{n}f(g(x))}{\mathrm {d}x^{n}}=\sum_{k=1}^{n}f^{(k)} \bigl(g(x)\bigr) \mathrm {B}_{n,k} \bigl(g'(x),g''(x), \ldots,g^{(n-k+1)}(x) \bigr). $$
(2.1)

Lemma 2.3

([12], p.135)

For \(n\ge k\ge0\), we have
$$ \mathrm{B}_{n,k}(1,1,\ldots,1)=S(n,k) $$
(2.2)
and
$$ \mathrm{B}_{n,k} \bigl(abx_{1},ab^{2}x_{2}, \ldots,ab^{n-k+1}x_{n-k+1} \bigr) =a^{k}b^{n} \mathrm{B}_{n,k}(x_{1},x_{n},\ldots,x_{n-k+1}), $$
(2.3)
where a and b are any complex numbers.

Lemma 2.4

([5], Theorem 2.1)

For \(n\ge k\ge1\), the Bell polynomials of the second kind \(\mathrm {B}_{n,k}\) satisfy
$$ \mathrm{B}_{n,k} \biggl(0,1,0,\ldots,\frac{1+(-1)^{n-k+1}}{2} \biggr) =\frac{1}{2^{k}k!}\sum_{\ell=0}^{2k}(-1)^{\ell} \binom{2k}{\ell }(k-\ell)^{n}. $$
(2.4)

Lemma 2.5

([5], Theorem 4.1 and [13], Theorem 3.1)

For \(n\ge k\ge0\), the Bell polynomials of the second kind \(\mathrm {B}_{n,k}\) satisfy
$$ \mathrm{B}_{n,k}(x,1,0,\ldots,0) =\frac{(n-k)!}{2^{n-k}} \binom{n}{k}\binom{k}{n-k}x^{2k-n}. $$
(2.5)

Lemma 2.6

([7], Theorem 1.2)

For \(n\ge k\ge1\), the Bell polynomials of the second kind \(\mathrm {B}_{n,k}\) satisfy
$$\begin{aligned}& \mathrm{B}_{n,k} \biggl(-\sin x,-\cos x,\sin x,\cos x,\ldots, -\sin \biggl[x+(n-k)\frac{\pi}{2} \biggr] \biggr) \\& \quad =(-1)^{k+\frac{1}{2} [n+\frac{1-(-1)^{n}}{2} ]}\frac{1}{k!}\cos^{k}x \sum _{\ell=1}^{k}(-1)^{\ell}\frac{1}{2^{\ell}} \binom{k}{\ell}\frac{1}{\cos^{\ell}x} \\& \qquad {}\times\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{n} \sin \biggl[(2q-\ell)x+\frac{1+(-1)^{n}}{2} \frac{\pi}{2} \biggr]. \end{aligned}$$

3 Proofs of main results

We now start out to prove our main results.

Proof of Theorem 1.1

Applying Lemma 2.1 to \(u(z)=1\) and \(v(z)=\cos z\) gives
$$\begin{aligned} (\sec z)^{(n)}&=(-1)^{n}\frac{| [\delta_{i1}]_{1\le i\le n+1} \ A_{(n+1)\times n}|_{(n+1)\times(n+1)}}{ \cos^{n+1}z} \\ &=(-1)^{n}\biggl\vert \binom{i-1}{j-1}\cos \biggl(z+(i-j) \frac{\pi}{2} \biggr) \biggr\vert _{2\le i\le n+1, 1\le j\le n} \\ &=(-1)^{n}\biggl\vert \binom{i}{j-1}\cos \biggl(z+(i-j+1) \frac{\pi}{2} \biggr)\biggr\vert _{n\times n}, \end{aligned}$$
where
$$ A_{(n+1)\times n}= \biggl[\binom{i-1}{j-1}\cos \biggl(z+(i-j) \frac{\pi}{2} \biggr)\biggr] _{1\le i\le n+1, 1\le j\le n} $$
and
$$ \delta_{ij}= \textstyle\begin{cases} 1, & i=j, \\ 0 & i\ne j \end{cases} $$
is the Kronecker delta. Consequently, by taking the limit \(z\to0\), we find
$$ \lim_{z\to0}(\sec z)^{(n)} =(-1)^{n}\biggl\vert \binom{i}{j-1}\cos \biggl((i-j+1)\frac{\pi}{2} \biggr) \biggr\vert _{n\times n} $$
and, by (1.1) and (1.3),
$$ (-1)^{k}E_{2k}=S_{k}=\lim_{z\to0}( \sec z)^{(2k)} = \biggl\vert \binom{i}{j-1}\cos \biggl((i-j+1) \frac{\pi}{2} \biggr) \biggr\vert _{(2k)\times(2k)}. $$
The proof of Theorem 1.1 is complete. □

Proof of Theorem 1.2

By (2.1) applied to \(f(u)=\frac{1}{u}\) and \(u=g(z)=\cos z\) and by Lemma 2.6, we have
$$\begin{aligned} (\sec z)^{(n)} =&\sum_{k=1}^{n} \biggl(\frac{1}{u} \biggr)^{(k)} \mathrm{B}_{n,k} \biggl(-\sin z,-\cos z,\sin z,\ldots, -\sin \biggl[z+(n-k)\frac{\pi}{2} \biggr] \biggr) \\ =&\sum_{k=1}^{n}\frac{(-1)^{k}k!}{u^{k+1}} \mathrm{B}_{n,k} \biggl(-\sin z,-\cos z,\sin z,\cos z,\ldots, -\sin \biggl[z+(n-k)\frac{\pi}{2} \biggr] \biggr) \\ =&\sum_{k=1}^{n}\frac{(-1)^{k}k!}{(\cos z)^{k+1}} \mathrm{B}_{n,k} \biggl(-\sin z,-\cos z,\sin z,\cos z,\ldots, -\sin \biggl[z+(n-k)\frac{\pi}{2} \biggr] \biggr) \\ =&\sum_{k=1}^{n}\frac{(-1)^{k}k!}{(\cos z)^{k+1}} (-1)^{k+\frac{1}{2} [n+\frac{1-(-1)^{n}}{2} ]}\frac{1}{k!}\cos^{k}z \sum _{\ell=1}^{k}(-1)^{\ell}\frac{1}{2^{\ell}} \binom{k}{\ell}\frac{1}{\cos^{\ell}z} \\ &{} \times\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{n} \sin \biggl[(2q-\ell)z+\frac{1+(-1)^{n}}{2} \frac{\pi}{2} \biggr] \\ \to&(-1)^{\frac{1}{2} [n+\frac{1-(-1)^{n}}{2} ]}\sum_{k=1}^{n} \sum _{\ell=1}^{k}(-1)^{\ell}\frac{1}{2^{\ell}}\binom{k}{\ell} \\ &{}\times\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{n} \sin \biggl[\frac {1+(-1)^{n}}{2} \frac{\pi}{2} \biggr] \end{aligned}$$
as \(z\to0\). Hence,
$$\begin{aligned} (-1)^{m}E_{2m} =&(-1)^{\frac{1}{2} [2m+\frac{1-(-1)^{2m}}{2} ]}\sum _{k=1}^{2m} \sum_{\ell=1}^{k}(-1)^{\ell}\frac{1}{2^{\ell}}\binom{k}{\ell} \\ &{} \times\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2m} \sin \biggl[\frac{1+(-1)^{2m}}{2} \frac{\pi}{2} \biggr] \\ =&(-1)^{m}\sum_{k=1}^{2m} \sum _{\ell=1}^{k}(-1)^{\ell}\frac{1}{2^{\ell}}\binom{k}{\ell} \sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2m}. \end{aligned}$$
Consequently, by interchanging the first two sums, we get
$$\begin{aligned} E_{2m}&=\sum_{\ell=1}^{2m} \sum _{k=\ell}^{2m}(-1)^{\ell}\frac{1}{2^{\ell}}\binom{k}{\ell} \sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell )^{2m} \\ &=\sum_{\ell=1}^{2m} (-1)^{\ell}\frac{1}{2^{\ell}} \Biggl[\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2m} \Biggr] \Biggl[\sum _{k=\ell}^{2m}\binom{k}{\ell} \Biggr] \\ &=\sum_{\ell=1}^{2m} (-1)^{\ell}\frac{1}{2^{\ell}} \Biggl[\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2m} \Biggr] \frac{2 m-\ell +1}{\ell+1} \binom{2 m+1}{\ell} \\ &=\sum_{\ell=1}^{2m} (-1)^{\ell}\frac{1}{2^{\ell}} \Biggl[\sum_{q=0}^{\ell}\binom{\ell}{q}(2q-\ell)^{2m} \Biggr] \frac{2 m+1}{\ell +1} \binom{2 m}{\ell} \end{aligned}$$
which may be rearranged as (1.4). The proof of Theorem 1.2 is complete. □

Proof of Theorem 1.3

Let \(f(u)=\frac{2u}{u^{2}+1}\) and \(u=g(z)=e^{z}\). Then, by (2.1), (2.3), and (2.2) in sequence,
$$\begin{aligned} \frac{\mathrm {d}^{n}}{\mathrm {d}z^{n}} \biggl(\frac{2e^{z}}{e^{2z}+1} \biggr) &=\sum _{k=1}^{n}\frac{\mathrm {d}^{k}}{\mathrm {d}u^{k}} \biggl(\frac{2u}{u^{2}+1} \biggr)\mathrm{B}_{n,k}\bigl(e^{z},e^{z}, \ldots,e^{z}\bigr) \\ &=\sum_{k=1}^{n}\frac{\mathrm {d}^{k+1}\ln(u^{2}+1)}{\mathrm {d}u^{k+1}}e^{kz} \mathrm {B}_{n,k}(1,1,\ldots,1) \\ &=\sum_{k=1}^{n}S(n,k)e^{kz} \frac{\mathrm {d}^{k+1}\ln(u^{2}+1)}{\mathrm {d}u^{k+1}}, \end{aligned}$$
where, by applying \(f(v)=\ln v\) and \(v=g(u)=u^{2}+1\) in (2.1) and making use of (2.3) and (2.5), we have
$$\begin{aligned} \frac{\mathrm {d}^{k+1}\ln(u^{2}+1)}{\mathrm {d}u^{k+1}} &=\sum_{\ell=1}^{k+1}(\ln v)^{(\ell)}\mathrm{B}_{k+1,\ell }(2u,2,0,\ldots,0) \\ &=\sum_{\ell=1}^{k+1}(-1)^{\ell-1} \frac{(\ell-1)!}{v^{\ell}}2^{\ell}\mathrm{B}_{k+1,\ell}(u,1,0,\ldots,0) \\ &=\sum_{\ell=1}^{k+1}(-1)^{\ell-1} \frac{(\ell-1)!}{(u^{2}+1)^{\ell}}2^{\ell}\frac{(k+1-\ell)!}{2^{k+1-\ell}}\binom{k+1}{\ell} \binom{\ell }{k+1-\ell}u^{2\ell-k-1} \\ &=\sum_{\ell=1}^{k+1}(-1)^{\ell-1} \frac{(\ell-1)!(k+1-\ell )!}{2^{k+1-2\ell}} \binom{k+1}{\ell}\binom{\ell}{k+1-\ell} \frac {e^{(2\ell-k-1)z}}{(e^{2z}+1)^{\ell}}. \end{aligned}$$
Consequently, we obtain
$$\begin{aligned} \frac{\mathrm {d}^{n}}{\mathrm {d}z^{n}} \biggl(\frac{2e^{z}}{e^{2z}+1} \biggr) =&\sum _{k=1}^{n}S(n,k)\sum_{\ell=1}^{k+1}(-1)^{\ell-1} \frac{(\ell -1)!(k+1-\ell)!}{2^{k+1-2\ell}} \\ &{} \times\binom{k+1}{\ell}\binom{\ell}{k+1-\ell} \frac {e^{(2\ell-1)z}}{(e^{2z}+1)^{\ell}}. \end{aligned}$$
Further taking the limit \(z\to0\) yields
$$\begin{aligned} E_{n}&=\lim_{z\to0}\frac{\mathrm {d}^{n}}{\mathrm {d}z^{n}} \biggl( \frac {2e^{z}}{e^{2z}+1} \biggr) \\ &=\sum_{k=1}^{n}S(n,k)\sum _{\ell=1}^{k+1}(-1)^{\ell-1}\frac{(\ell -1)!(k+1-\ell)!}{2^{k+1-\ell}} \binom{k+1}{\ell}\binom{\ell }{k+1-\ell} \\ &=S(n,1)+\sum_{k=1}^{n}S(n,k) \sum _{\ell=2}^{k+1}(-1)^{\ell-1}\frac {(\ell-1)!(k+1-\ell)!}{2^{k+1-\ell}} \binom{k+1}{\ell}\binom{\ell }{k+1-\ell} \\ &=1+\sum_{k=1}^{n}S(n,k) \sum _{\ell=1}^{k}(-1)^{\ell}\frac{\ell !(k-\ell)!}{2^{k-\ell}} \binom{k+1}{\ell+1}\binom{\ell+1}{k-\ell} \\ &=1+\sum_{k=1}^{n}\frac{1}{2^{k}} \Biggl[\sum_{\ell=1}^{k}(-1)^{\ell}\ell !(k-\ell)! 2^{\ell} \binom{k+1}{\ell+1}\binom{\ell+1}{k-\ell } \Biggr]S(n,k) \end{aligned}$$
and, by interchanging two sums in the above line, we get
$$\begin{aligned} E_{n}&=1+\sum_{\ell=1}^{n} \sum _{k=\ell}^{n}(-1)^{\ell} \frac{\ell !(k-\ell)!}{2^{k-\ell}} \binom{k+1}{\ell+1}\binom{\ell+1}{k-\ell }S(n,k) \\ &=1+\sum_{\ell=1}^{n}(-1)^{\ell} \Biggl[\sum_{k=0}^{n-\ell}\frac {k!}{2^{k}} \binom{k+\ell+1}{\ell+1}\binom{\ell+1}{k}S(n,k+\ell ) \Biggr]\ell!. \end{aligned}$$
As a result of further simplifying, formulas (1.5) and (1.6) follow. The proof of Theorem 1.3 is complete. □

Proof of Theorem 1.4

Formula (1.7) was ever established in [5], Theorem 3.1. We now give a different proof for it.

Applying (2.1) to the functions \(f(u)=\frac{1}{u}\) and \(u=g(z)=\cosh z\) and making use of formula (2.4) yield
$$\begin{aligned} \frac{\mathrm {d}^{n}}{\mathrm {d}z^{n}} \biggl(\frac{1}{\cosh z} \biggr)& =\sum _{k=1}^{n} \biggl(\frac{1}{u} \biggr)^{(k)}\mathrm{B}_{n,k}(\sinh z,\cosh z,\sinh z,\ldots) \\ &=\sum_{k=1}^{n}\frac{(-1)^{k}k!}{u^{k+1}} \mathrm{B}_{n,k}(\sinh z,\cosh z,\sinh z,\ldots) \\ &=\sum_{k=1}^{n}\frac{(-1)^{k}k!}{(\cosh z)^{k+1}} \mathrm{B}_{n,k}(\sinh z,\cosh z,\sinh z,\ldots) \\ &\to\sum_{k=1}^{n}(-1)^{k}k! \mathrm{B}_{n,k} \biggl(0,1,0,\ldots,\frac {1+(-1)^{n-k+1}}{2} \biggr), \quad z \to0 \\ &=\sum_{k=1}^{n}(-1)^{k}k! \frac{1}{2^{k}k!}\sum_{\ell=0}^{2k}(-1)^{\ell } \binom{2k}{\ell}(k-\ell)^{n} \\ &=\sum_{k=1}^{n}(-1)^{k} \frac{1}{2^{k}}\sum_{\ell=0}^{2k}(-1)^{\ell} \binom {2k}{\ell}(k-\ell)^{n}. \end{aligned}$$
By virtue of (1.2), Theorem 1.4 follows immediately. □

Remark 3.1

This paper is a slightly corrected and revised version of the preprint [14].

Declarations

Acknowledgements

The first author was partially supported by the NNSF of China under Grant No. 51274086, by the Ministry of Education Doctoral Foundation of China - Priority Areas under Grant No. 20124116130001, by the Basic and Frontier Research Project in Henan Province of China under Grant No. 122300410115, and by the Doctoral Foundation at Henan Polytechnic University in China under Grant No. B2014-003. The second author was partially supported by the NNSF of China under Grant No. 11361038. The authors thank anonymous referees for their careful corrections to the original version of this paper.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematics and Informatics, Henan Polytechnic University
(2)
School of Energy Science and Engineering, Henan Polytechnic University
(3)
College of Mathematics, Inner Mongolia University for Nationalities
(4)
Department of Mathematics, College of Science, Tianjin Polytechnic University

References

  1. Abramowitz, M, Stegun, IA (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th edn. Applied Mathematics Series, vol. 55. National Bureau of Standards, Washington (1972) MATHGoogle Scholar
  2. Temme, NM: Special Functions: An Introduction to Classical Functions of Mathematical Physics. A Wiley-Interscience Publication. Wiley, New York (1996). doi:10.1002/9781118032572 MATHView ArticleGoogle Scholar
  3. Brent, RP, Harvey, D: Fast computation of Bernoulli, tangent and secant numbers. In: Computational and Analytical Mathematics. Springer Proc. Math. Stat., vol. 50, pp. 127-142. Springer, New York (2013). doi:10.1007/978-1-4614-7621-4_8 View ArticleGoogle Scholar
  4. Guo, B-N, Qi, F: Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind. J. Comput. Appl. Math. 272, 251-257 (2014). doi:10.1016/j.cam.2014.05.018 MATHMathSciNetView ArticleGoogle Scholar
  5. Guo, B-N, Qi, F: Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers. ResearchGate Technical Report (2015). doi:10.13140/2.1.3794.8808
  6. Kim, DS, Kim, T, Kim, YH, Lee, SH: Some arithmetic properties of Bernoulli and Euler numbers. Adv. Stud. Contemp. Math. (Kyungshang) 22(4), 467-480 (2012) MATHMathSciNetGoogle Scholar
  7. Qi, F: Derivatives of tangent function and tangent numbers. Appl. Math. Comput. (2015, in press). arXiv:1202.1205v3
  8. Malenfant, J: Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers (2011). arXiv:1103.1585
  9. Vella, DC: Explicit formulas for Bernoulli and Euler numbers. Integers 8, A01 (2008) MathSciNetGoogle Scholar
  10. Yakubovich, S: Certain identities, connection and explicit formulas for the Bernoulli, Euler numbers and Riemann zeta-values (2014). arXiv:1406.5345
  11. Bourbaki, N: Functions of a Real Variable, Elementary Theory: Elements of Mathematics. Springer, Berlin (2004). doi:10.1007/978-3-642-59315-4; Translated from the 1976 French original by Philip Spain. View ArticleGoogle Scholar
  12. Comtet, L: Advanced Combinatorics: The Art of Finite and Infinite Expansions, revised and enlarged edition. Reidel, Dordrecht (1974) MATHView ArticleGoogle Scholar
  13. Qi, F, Zheng, M-M: Explicit expressions for a family of the Bell polynomials and applications. Appl. Math. Comput. 258, 597-607 (2015). doi:10.1016/j.amc.2015.02.027 MathSciNetView ArticleGoogle Scholar
  14. Qi, F, Wei, C-F: Several closed expressions for the Euler numbers. ResearchGate Technical Report (2015). doi:10.13140/2.1.3474.7688

Copyright

© Wei and Qi 2015