SEVERAL CLOSED EXPRESSIONS FOR THE EULER NUMBERS 1 ...

SEVERAL CLOSED EXPRESSIONS FOR THE EULER NUMBERS FENG QI AND CHUN-FU WEI

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.

1. Introduction and main results It is well known [1, p. 75, 4.3.69] that the secant function sec z may be expanded at z = 0 into the power series ∞ X z 2k π sec z = (−1)k E2k , |z| < , (1.1) (2k)! 2 k=0

where Ek are called in number theory the Euler numbers which may also be defined [11, p. 15] by ∞ ∞ X X 2ez π zk z 2k 1 = 2z = Ek = E2k , |z| < . (1.2) cosh z e +1 k! (2k)! 2 k=0

k=0

In number theory, the numbers Sk = (−1)k E2k (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 Sk for k = 1, 2, 3, 4 are 1, 5, 61, 1385. The first few Euler numbers E2k for 0 ≤ k ≤ 9 are E0 = 1, E12 = 2702765,

E2 = −1,

E4 = 5,

E14 = −199360981,

E6 = −61,

E8 = 1385,

E16 = 19391512145,

E10 = −50521,

E18 = −2404879675441.

It is a classical topic to find closed expressions for the Euler numbers E2k and the tangent number Sk . 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 have been plenty of literature such as [1, 5, 6, 7, 8, 10, 11, 12, 13] and closely related references therein. In this paper, we will establish several closed expressions for the Euler numbers E2k 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 [11, p. 20] by ∞ X (ex − 1)k xn = S(n, k) , k ∈ N, (1.4) k! n! n=k

2010 Mathematics Subject Classification. Primary 11B68; Secondary 11B73, 33B10. Key words and phrases. Euler number; closed expression; determinant; Stirling number of the second kind; double sum. This paper was typeset using AMS-LATEX. 1

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F. QI AND C.-F. WEI

may be computed [11, p. 21] by the closed expression   m m k 1 X (−1)m−` ` , S(k, m) = m! `

1 ≤ m ≤ k,

(1.5)

`=1

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 ∈ N,     i π cos (i − j + 1) , E2k = (−1) j−1 2 (2k)×(2k)   is the determinant of a matrix ci,j k×k of elements ci,j and order k. k

where ci,j k×k

(1.6)

Theorem 1.2. For k ∈ N, E2k = (2k + 1)

2k X

(−1)`

`=1

 X `   1 2k ` (2q − `)2k . 2` (` + 1) ` q=0 q

(1.7)

  k ` X `+1 ` 2 S(n, k) (−1) `+1 k−`

(1.8)

Theorem 1.3. For n ∈ N, En = 1 +

n X (k + 1)!

2k

k=1

`=1

and En = 1 +

n X

(−1)`

  n−` 1 X (k + ` + 1)! ` + 1 S(n, k + `). `+1 2k k

(1.9)

k=0

`=1

Theorem 1.4. For k ∈ N, E2k =

2k X i=1

(−1)i

  2i 1 X ` 2i (−1) (i − `)2k . ` 2i

(1.10)

`=0

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) 6= 0 be differentiable functions, let Un+1,1 be an (n + 1) × 1 matrix whose elements
uk,1 = u(k−1) (x) for 1 ≤ k ≤ n + 1, let Vn+1,n be an (n + 1) × n i−1 (i−j) matrix whose elements vi,j = j−1 v (x) for 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n, and let |Wn+1,n+1 |   denote the determinant of the (n + 1) × (n + 1) matrix Wn+1,n+1 = Un+1,1 Vn+1,n . Then the n-th derivative of the ratio u(x) v(x) may be computed by   dn u |Wn+1,n+1 | = (−1)n . (2.1) n dx v v n+1 Proof. This is a reformulation of [2, p. 40, Exercise 5)].



Lemma 2.2 (Fa` a di Bruno formula [4, p. 139, Theorem C]). For n ∈ N, the n-th derivative of a composite function f (g(x)) may be computed in terms of the Bell polynomials of the second kind Bn,k by n
dn f (g(x)) X (k) = f (g(x))Bn,k g 0 (x), g 00 (x), . . . , g (n−k+1) (x) , (2.2) n dx k=1

SEVERAL CLOSED EXPRESSIONS FOR THE EULER NUMBERS

3

where the Bell polynomials of the second kind Bn,k are also called as the partial Bell polynomials and may be defined [4, p. 134, Theorem A] by n−k+1 X Y  x q  `q n! , n ≥ k ≥ 0. (2.3) Bn,k (x1 , x2 , . . . , xn−k+1 ) = Qn−k+1 q! `q ! q=1 q=1 1≤q≤n,` Pn q ∈{0}∪N i`q =n Pq=1 n q=1 `q =k

Lemma 2.3 ([4, p. 135]). For n ≥ k ≥ 0, we have Bn,k (1, 1, . . . , 1) = S(n, k)

(2.4)


Bn,k abx1 , ab2 x2 , . . . , abn−k+1 xn−k+1 = ak bn Bn,k (x1 , xn , . . . , xn−k+1 ),

(2.5)

and where a and b are any complex numbers. Lemma 2.4 ([6, Theorem 2.1]). For n ≥ k ≥ 1, the Bell polynomials of the second kind Bn,k satisfy     2k 1 X 1 + (−1)n−k+1 ` 2k = k (−1) (k − `)n . (2.6) Bn,k 0, 1, 0, . . . , 2 2 k! ` `=0

Lemma 2.5 ([6, Theoem 4.1] and [9, Theorem 3.1]). For n ≥ k ≥ 0, the Bell polynomials of the second kind Bn,k satisfy    k (n − k)! n x2k−n . (2.7) Bn,k (x, 1, 0, . . . , 0) = n−k 2n−k k Lemma 2.6 ([8, Theorem 1.2]). For n ≥ k ≥ 1, the Bell polynomials of the second kind Bn,k satisfy    π Bn,k − sin x, − cos x, sin x, cos x, . . . , − sin x + (n − k) 2    1−(−1)n  k X 1 1 1 k 1 = (−1)k+ 2 n+ 2 cosk x (−1)` ` k! 2 ` cos` x `=1     ` X ` 1 + (−1)n π × (2q − `)n sin (2q − `)x + . (2.8) 2 2 q q=0 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 h  

i i−1 δ π cos x + (i − j) i,1 1≤i≤n+1 j−1 2 1≤i≤n+1,1≤j≤n (n+1)×(n+1) (n) n (sec z) = (−1) n+1 z cos     π n i−1 = (−1) cos x + (i − j) j−1 2 2≤i≤n+1,1≤j≤n     i π = (−1)n cos x + (i − j + 1) , j−1 2 n×n

4

F. QI AND C.-F. WEI

where ( δi,j =

1, i = j 0 i 6= j

is the Kronecker delta. Consequently, by taking the limit z → 0, we find     i π (n) n lim (sec z) = (−1) cos (i − j + 1) z→0 j−1 2 n×n and, by (1.1) and (1.3), k

(2k)

(−1) E2k = Sk = lim (sec z) z→0

    i π cos (i − j + 1) = j−1 2

. (2k)×(2k)

The proof of Theorem 1.1 is complete.



Proof of Theorem 1.2. By (2.2) applied to f (u) = u1 and u = g(z) = cos z and by Lemma 2.6, we have    n  (k) X π 1 (sec z)(n) = Bn,k − sin z, − cos z, sin z, cos z, . . . , − sin z + (n − k) u 2 k=1    n X (−1)k k! π = Bn,k − sin z, − cos z, sin z, cos z, . . . , − sin z + (n − k) uk+1 2 k=1    n X π (−1)k k! B − sin z, − cos z, sin z, cos z, . . . , − sin z + (n − k) = n,k (cos z)k+1 2 k=1    1−(−1)n  n k X X (−1)k k! 1 k 1 1 k+ 12 n+ k 2 = (−1) cos z (−1)` ` k+1 (cos z) k! 2 ` cos` z k=1 `=1   `   X ` 1 + (−1)n π n × (2q − `) sin (2q − `)z + q 2 2 q=0 1 2

→ (−1)



n+

1−(−1)n 2

 X   X n X k `   ` 1 + (−1)n π ` 1 k n (−1) ` (2q − `) sin 2 ` q=0 q 2 2 k=1 `=1

as z → 0. Hence, 1

(−1)m E2m = (−1) 2



2m+

1−(−1)2m 2

   `    X 2m X k 1 + (−1)2m π 1 k X ` (2q − `)2m sin (−1)` ` 2 ` q=0 q 2 2 k=1 `=1

= (−1)m

2m X k X

(−1)`

k=1 `=1

  `   1 k X ` (2q − `)2m . 2` ` q=0 q

Consequently, by interchanging the first two sums,  X 2m X 2m `   X ` ` 1 k E2m = (−1) ` (2q − `)2m 2 ` q=0 q `=1 k=` " `   #" 2m  # 2m X X ` X k 1 = (−1)` ` (2q − `)2m 2 q=0 q ` `=1

k=`

SEVERAL CLOSED EXPRESSIONS FOR THE EULER NUMBERS

5

" `   #   1 X ` 2m 2m − ` + 1 2m + 1 = (2q − `) (−1) ` 2 q=0 q `+1 ` `=1 " #   `   2m X 2m + 1 2m 1 X ` = (2q − `)2m (−1)` ` 2 q=0 q `+1 ` 2m X

`

`=1

which may be rearranged as (1.7). The proof of Theorem 1.2 is complete.



z Proof of Theorem 1.3. Let f (u) = u22u +1 and u = g(z) = e . Then, by (2.2), (2.5), and (2.4) in sequence,    X  n 2ez 2u dn dk = Bn,k (ez , ez , . . . , ez ) d xn e2z + 1 d xk u2 + 1

= =

k=1 n X k=1 n X

k=1

dk+1 ln(u2 + 1) kz e Bn,k (1, 1, . . . , 1) d xk+1 S(n, k)ekz

dk+1 ln(u2 + 1) , d xk+1

where, by applying f (v) = ln v and v = g(u) = u2 + 1 in (2.2) and making use of (2.5) and (2.7), k+1

dk+1 ln(u2 + 1) X (ln v)(`) Bk+1,` (2u, 2, 0, . . . , 0) = d xk+1 `=1

k+1 X

(` − 1)! ` 2 Bk+1,` (u, 1, 0, . . . , 0) v` `=1    k+1 X ` `−1 (` − 1)! ` (k + 1 − `)! k + 1 = (−1) 2 u2`−k−1 (u2 + 1)` 2k+1−` ` k+1−` `=1    (2`−k−1)z k+1 X ` e `−1 (` − 1)!(k + 1 − `)! k + 1 . = (−1) 2k+1−2` ` k + 1 − ` (e2z + 1)`

=

(−1)`−1

`=1

Consequently, we obtain   X    (2`−1)z n k+1 X 2ez ` e dn `−1 (` − 1)!(k + 1 − `)! k + 1 = S(n, k) (−1) . k + 1 − ` (e2z + 1)` d xn e2z + 1 2k+1−2` ` k=1

`=1

Further taking the limit z → 0 yields   dn 2ez En = lim x→0 d xn e2z + 1 n X

k+1 X

   (` − 1)!(k + 1 − `)! k + 1 ` 2k+1−` ` k+1−` k=1 `=1    n k+1 X X (` − 1)!(k + 1 − `)! k + 1 ` = S(n, 1) + S(n, k) (−1)`−1 2k+1−` ` k+1−` k=1 `=2    n k X X `!(k − `)! k + 1 ` + 1 =1+ S(n, k) (−1)` 2k−` `+1 k−` =

S(n, k)

k=1

(−1)`−1

`=1

6

F. QI AND C.-F. WEI

" k   # n X `+1 1 X ` ` k+1 =1+ (−1) `!(k − `)!2 S(n, k) 2k `+1 k−` k=1

`=1

and, by interchanging two sums in the above line,    n X n X `+1 ` `!(k − `)! k + 1 S(n, k) En = 1 + (−1) `+1 k−` 2k−` `=1 k=` "n−`  # n X X k! k + ` + 1` + 1 ` =1+ (−1) S(n, k + `) `!. `+1 k 2k `=1

k=0

As a results of further simplifying, the formulas (1.8) and (1.9) follow. The proof of Theorem 1.3 is complete.  Proof of Theorem 1.4. The formula (1.10) was ever established in [6, Theorem 3.1]. We now give a different proof for it. Applying (2.2) to the functions f (u) = u1 and u = g(z) = cosh z and making use of the formula (2.6) yield   X n  (k) 1 dn 1 = Bn,k (sinh z, cosh z, sinh z, . . . ) d z n cosh z u =

k=1 n X

k=1 n X

(−1)k k! Bn,k (sinh z, cosh z, sinh z, . . . ) uk+1

(−1)k k! Bn,k (sinh z, cosh z, sinh z, . . . ) (cosh z)k+1 k=1   n X 1 + (−1)n−k+1 k → (−1) k!Bn,k 0, 1, 0, . . . , , 2 k=1   2k n X 1 X ` 2k k (−1) (k − `)n (−1) k! k = ` 2 k! `=0 k=1   n 2k X X 1 2k (−1)k k = (k − `)n . (−1)` ` 2 =

k=1

z→0

`=0

By virtue of (1.2), Theorem 1.4 follows immediately.



Acknowledgements. The first author was partially supported by the NNSF of China under Grant No. 11361038. The second 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. References [1] M. Abramowitz and I. A. Stegun (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series 55, 9th printing, Washington, 1970. [2] N. Bourbaki, Functions of a Real Variable, Elementary Theory, Translated from the 1976 French original by Philip Spain. Elements of Mathematics (Berlin). Springer-Verlag, Berlin, 2004; Available online at http: //dx.doi.org/10.1007/978-3-642-59315-4.

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[3] R. P. Brent and D. Harvey, Fast computation of Bernoulli, tangent and secant numbers, Computational and Analytical Mathematics, 127–142, Springer Proc. Math. Stat., 50, Springer, New York, 2013; Avaialble online at http://dx.doi.org/10.1007/978-1-4614-7621-4_8. [4] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. [5] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math. 272 (2014), 251–257; Available online at http://dx.doi.org/10. 1016/j.cam.2014.05.018. [6] B.-N. Guo and F. Qi, Explicit formulas for special values of the Bell polynomials of the second kind and the Euler numbers, ResearchGate Technical Report, available online at http://dx.doi.org/10.13140/2.1. 3794.8808. [7] D. S. Kim, T. Kim, Y. H. Kim, and S. H. Lee, Some arithmetic properties of Bernoulli and Euler numbers, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 4, 467–480. [8] F. Qi, Derivatives of tangent function and tangent numbers, available online at http://arxiv.org/abs/ 1202.1205v3. [9] F. Qi and M.-M. Zheng, Explicit expressions for a family of the Bell polynomials and applications, Appl. Math. Comput. (2015), in press; Available online at http://dx.doi.org/10.1016/j.amc.2015.02.027. [10] J. Malenfant, Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers, available online at http://arxiv.org/abs/1103.1585. [11] N. M. Temme, Special Functions: An Introduction to Classical Functions of Mathematical Physics, Wiley 1996. [12] D. C. Vella, Explicit formulas for Bernoulli and Euler numbers, Integers 8 (2008), A01, 7 pp. [13] S. Yakubovich, Certain identities, connection and explicit formulas for the Bernoulli, Euler numbers and Riemann zeta-values, available online at http://arxiv.org/abs/1406.5345. (F. Qi) College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China E-mail address: [email protected], [email protected], [email protected] URL: http://qifeng618.wordpress.com (C.-F. Wei) School of Mathematics and Informatics & School of Energy Science and Engineering, Henan Polytechnic University, Jiaozuo City, Henan Province, 454003, China E-mail address: [email protected], [email protected]