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Two identities and closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind

  • Xue-Yan Chen , Lan Wu , Dongkyu Lim EMAIL logo and Feng Qi EMAIL logo
Published/Copyright: November 16, 2022
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

In this article, the authors present two identities involving products of the Bernoulli numbers, provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers in terms of central factorial numbers of the second kind, and supply simple proofs of series expansions of (hyperbolic) cosecant and cotangent functions.

Keywords: Bernoulli number; identity; product; cosecant; cotangent; hyperbolic cosecant; hyperbolic cotangent; series expansion; central factorial number of the second kind; closed-form formula
MSC 2010: Primary 11B68; Secondary 11B73; 33B10

1 Introduction

The Bernoulli numbers B k for k ≥ 0 are generated in [1, p. 3] by

z e z − 1 = ∑ k = 0 ∞ B k z k k ! = 1 − z 2 + ∑ k = 1 ∞ B 2 k z 2 k ( 2 k ) ! , ∣ z ∣ < 2 π .

More generally, the Bernoulli polynomials B k ( x ) for k ≥ 0 are defined in [1, p. 3] by the exponential generating function

z e x z e z − 1 = ∑ k = 0 ∞ B k ( x ) z k k ! , ∣ z ∣ < 2 π

for x ∈ R . It is clear that B k = B k ( 0 ) for k ≥ 0 . The Bernoulli numbers and polynomials B k and B k ( x ) for k ≥ 0 and x ∈ R are fundamental concepts in mathematics and have been applied extensively in mathematical sciences. The study of the Bernoulli numbers and polynomials B k and B k ( x ) has a long history, but there are still new conclusions, for example, those results in the articles [2,3,4], emerged in recent years.

The central factorial numbers of the second kind T ( n , k ) for n ≥ k ≥ 0 can be generated [5,6] by

(1.1) 1 k ! 2 sinh x 2 k = ∑ n = k ∞ T ( n , k ) x n n ! .

In [7, Chapter 6, equation (26)], it was established that

(1.2) T ( n , k ) = 1 k ! ∑ ℓ = 0 k ( − 1 ) ℓ k ℓ k 2 − ℓ n

with T ( 0 , 0 ) = 1 . See also [5, Proposition 2.4, (xii)] and [8,9].

In this article, we present two identities involving the products B 2 j B 2 n − 2 j of the Bernoulli numbers B n , provide four alternative proofs for these two identities, derive two closed-form formulas for the Bernoulli numbers B 2 m in terms of central factorial numbers of the second kind T ( n , k ) , and supply simple proofs of series expansions of the (hyperbolic) cosecant and cotangent functions.

2 The first identity and its two proofs

At the site https://www.researchgate.net/post/How_can_I_compute_the_symbolic_expression_of_the_Taylor_series_of_x2cscxcschx, Roudy El Haddad asked a question: How can I compute the symbolic expression of the Taylor series of x 2 csc x csch x ? This question is due to [10, p. 221, Example 4.1]. The answer to this question is included in the following theorem.

Theorem 2.1

For k ∈ N = { 1 , 2 , … } , we have

(2.1) ∑ j = 1 2 k ( − 1 ) j 4 k + 2 2 j ( 2 2 j − 1 − 1 ) ( 2 4 k − 2 j + 1 − 1 ) B 2 j B 4 k − 2 j + 2 = 0 .

For x ∈ ( − π , π ) , we have

(2.2) x 2 csc x csch x = 1 − 4 ∑ k = 1 ∞ ( 2 4 k − 1 − 1 ) B 4 k − ∑ j = 1 2 k − 1 ( − 1 ) j 4 k 2 j ( 2 2 j − 1 − 1 ) ( 2 4 k − 2 j − 1 − 1 ) B 2 j B 4 k − 2 j x 4 k ( 4 k ) ! = 1 + x 4 90 + 13 x 8 113400 + 4009 x 12 3405402000 + 13739 x 16 1136785104000 + ⋯ .

First proof

On page 42 in the handbook [11], the series expansions

(2.3) csc x = 1 x + ∑ k = 1 ∞ 2 ( 2 2 k − 1 − 1 ) ∣ B 2 k ∣ ( 2 k ) ! x 2 k − 1

and

(2.4) csch x = 1 x − ∑ k = 1 ∞ 2 ( 2 2 k − 1 − 1 ) B 2 k ( 2 k ) ! x 2 k − 1

for x ∈ ( − π , π ) are collected. Hence, by the Cauchy product of two infinite series in mathematical analysis [11, p. 19], we obtain

x 2 csc x csch x = ( x csc x ) ( x csch x ) = 1 + ∑ k = 1 ∞ 2 ( 2 2 k − 1 − 1 ) ∣ B 2 k ∣ ( 2 k ) ! x 2 k 1 − ∑ k = 1 ∞ 2 ( 2 2 k − 1 − 1 ) B 2 k ( 2 k ) ! x 2 k = 1 + 2 ∑ k = 2 ∞ ( 2 2 k − 1 − 1 ) ( ∣ B 2 k ∣ − B 2 k ) − 2 ∑ j = 1 k − 1 2 k 2 j ( 2 2 j − 1 − 1 ) ( 2 2 k − 2 j − 1 − 1 ) ∣ B 2 j ∣ B 2 k − 2 j x 2 k ( 2 k ) !

for x ∈ ( − π , π ) .

Since the function x csc x csch x is even on the interval ( − π 2 , π 2 ) , or say, the function f ( x ) = x 2 csc x csch x satisfies f ( x ) = f ( x i ) on ( − π , π ) , identity (2.1) follows readily.

Making use of identity (2.1) and the fact that

( − 1 ) k + 1 B 2 k > 0 , k ∈ N

listed in [12, p. 805, 23.1.15], we finally deduce

x 2 csc x csch x = 1 − 4 ∑ k = 1 ∞ ( 2 4 k − 1 − 1 ) B 4 k − ∑ j = 1 2 k − 1 ( − 1 ) j 4 k 2 j ( 2 2 j − 1 − 1 ) ( 2 4 k − 2 j − 1 − 1 ) B 2 j B 4 k − 2 j x 4 k ( 4 k ) ! = 1 + x 4 90 + 13 x 8 113400 + 4009 x 12 3405402000 + 13739 x 16 1136785104000 + ⋯

for x ∈ ( − π , π ) . The series expansion (2.2) is thus proved.

Identity (2.1) was also announced as a question at https://mathoverflow.net/q/419493/ for a reference or a proof of it. At the site https://mathoverflow.net/a/419498/, Alapan Das (adas2001.suri@gmail.com, https://stackexchange.com/users/14939952/alapan-das) at West Bengal in India provided an alternative proof of identity (2.1). For completeness, we recite Das’ proof with slight modifications as follows.

Second proof

Let

f ( x ) = x e x − 1 + x 2 − 1 = x 2 coth x 2 − 1 .

The left-hand side of identity (2.1) can be broken into four parts as

2 4 k ∑ j = 1 2 k ( − 1 ) j 4 k + 2 2 j B 2 j B 4 k − 2 j + 2 − ∑ j = 1 2 k ( − 1 ) j 2 2 j − 1 4 k + 2 2 j B 2 j B 4 k − 2 j + 2 − ∑ j = 1 2 k ( − 1 ) j 2 4 k − 2 j + 1 4 k + 2 2 j B 2 j B 4 k − 2 j + 2 + ∑ j = 1 2 k ( − 1 ) j 4 k + 2 2 j B 2 j B 4 k − 2 j + 2 .

The first part is ( 4 k + 2 ) ! times the coefficient of x 4 k + 2 in the series expansion of the function f ( 2 x i ) f ( 2 x ) 4 at x = 0 . Similarly, we can compute other three parts and acquire that the total sum is ( 4 k + 2 ) ! times the coefficient of x 4 k + 2 in the series expansion of the function

F ( x ) = f ( 2 x i ) f ( 2 x ) 4 − f ( x i ) f ( 2 x ) + f ( 2 x i ) f ( x ) 2 + f ( x ) f ( x i )

at x = 0 . Since f ( x ) is an even function, we can see that F ( x ) = F ( x i ) . But this requires the coefficient of x 4 k + 2 in the series expansion of F ( x ) at x = 0 to be equal to zero. Identity (2.1) is thus proved.

3 The second identity and its two proofs

The second identity involving the products B 2 j B 2 k − 2 j of the Bernoulli numbers B k is stated in the following theorem.

Theorem 3.1

For k ≥ 2 , we have

(3.1) ∑ j = 1 k − 1 2 k 2 j ( 1 − 2 2 j − 1 − 2 2 k − 2 j − 1 ) B 2 j B 2 k − 2 j = ( 2 2 k − 1 ) B 2 k .

First proof

It is known that

1 sin x = csc x and 1 tan x = cot x .

Hence, making use of the series expansions (2.3) and (2.4) and considering the series expansions

(3.2) cot x = 1 x − ∑ k = 1 ∞ 2 2 k ∣ B 2 k ∣ ( 2 k ) ! x 2 k − 1

and

(3.3) coth x = 1 x + ∑ k = 1 ∞ 2 2 k B 2 k ( 2 k ) ! x 2 k − 1

for x ∈ ( − π , π ) , we can straightforwardly obtain

x tan x 2 = 1 − B 2 ( 2 x ) 2 + ∑ k = 2 ∞ ( − 1 ) k 2 B 2 k + ∑ j = 1 k − 1 2 k 2 j B 2 j B 2 k − 2 j ( 2 x ) 2 k ( 2 k ) ! = 1 − 2 x 2 3 + x 4 15 + 2 x 6 189 + x 8 675 + 2 x 10 10395 + 1382 x 12 58046625 + ⋯ , x tanh x 2 = 1 + B 2 ( 2 x ) 2 + ∑ k = 2 ∞ 2 B 2 k + ∑ j = 1 k − 1 2 k 2 j B 2 j B 2 k − 2 j ( 2 x ) 2 k ( 2 k ) ! = 1 + 2 x 2 3 + x 4 15 − 2 x 6 189 + x 8 675 − 2 x 10 10395 + 1382 x 12 58046625 − ⋯ , x sinh x 2 = 1 − 2 B 2 x 2 − 4 ∑ k = 2 ∞ ( 2 2 k − 1 − 1 ) B 2 k − ∑ j = 1 k − 1 2 k 2 j ( 2 2 j − 1 − 1 ) ( 2 2 k − 2 j − 1 − 1 ) B 2 j B 2 k − 2 j x 2 k ( 2 k ) ! = 1 − x 2 3 + x 4 15 − 2 x 6 189 + x 8 675 − 2 x 10 10395 + 1382 x 12 58046625 − ⋯ ,

and

(3.4) x sin x 2 = 1 + 2 B 2 x 2 − 4 ∑ k = 2 ∞ ( − 1 ) k ( 2 2 k − 1 − 1 ) B 2 k − ∑ j = 1 k − 1 2 k 2 j ( 2 2 j − 1 − 1 ) ( 2 2 k − 2 j − 1 − 1 ) B 2 j B 2 k − 2 j x 2 k ( 2 k ) ! = 1 + x 2 3 + x 4 15 + 2 x 6 189 + x 8 675 + 2 x 10 10395 + 1382 x 12 58046625 + ⋯ ,

for x ∈ ( − π , π ) . Further considering the relation

x tan x 2 + x 2 = x sin x 2

or

x tanh x 2 − x 2 = x sinh x 2

and comparing coefficients of the terms x 2 k , we figure out the identity (3.1).

Identity (3.1) was also posted as a question at https://mathoverflow.net/q/419528/for an alternative proof. At the website https://mathoverflow.net/a/419561/, an anonymous mathematician EFinat-S (https://stackexchange.com/users/10780809/efinat-s) gave a proof of identity (3.1). For comparison, we recite EFinat-S’ proof with slight modifications as follows.

Second proof

The Bernoulli polynomials B k ( x ) for k ≥ 0 satisfy the general identity

∑ i = 0 n n i B i ( x ) B n − i ( y ) = ( 1 − n ) B n ( x + y ) + n ( x + y − 1 ) B n − 1 ( x + y ) ,

which can be proved using the definition of B k ( x ) , simple manipulations, and comparing coefficients. See also [13, p. 595, 24.14.1]. Especially when x = y = 0 , it becomes

(3.5) ∑ i = 0 n n i B i B n − i = ( 1 − n ) B n − n B n − 1 .

See also [13, p. 595, 24.14.2]. Especially when x = y = 1 2 , it becomes

∑ i = 0 n n i B i 1 2 B n − i 1 2 = ( 1 − n ) B n ( 1 ) .

Since B i ( 1 2 ) = ( 2 1 − i − 1 ) B i , the left-hand side of the last equation is

∑ i = 0 n n i B i 1 2 B n − i 1 2 = ∑ i = 0 n n i ( 2 1 − i − 1 ) B i ( 2 1 − n + i − 1 ) B n − i = ∑ i = 0 n n i ( 1 − 2 1 − i − 2 1 − n + i + 2 2 − n ) B i B n − i = ∑ i = 0 n n i B i B n − i + 2 2 − n ∑ i = 0 n n i ( 1 − 2 n − i − 1 − 2 i − 1 ) B i B n − i .

Hence, we acquire

( 1 − n ) B n ( 1 ) = ( 1 − n ) B n − n B n − 1 + 2 2 − n ∑ i = 0 n n i ( 1 − 2 n − i − 1 − 2 i − 1 ) B i B n − i .

If n ≥ 4 is even, then B n ( 1 ) = B n and B n − 1 = 0 , and it follows that

0 = ∑ i = 0 n n i ( 1 − 2 n − i − 1 − 2 i − 1 ) B i B n − i = ( 1 − 2 n ) B n + ∑ i = 2 n − 2 n i ( 1 − 2 n − i − 1 − 2 i − 1 ) B i B n − i ,

which proves identity (3.1).

4 Two closed-form formulas for Bernoulli numbers

In this section, basing on Theorems 2.1 and 3.1 and their proofs, we derive two closed-form formulas for the Bernoulli numbers B 2 m in terms of central factorial numbers of the second kind T ( n , k ) .

Theorem 4.1

For m ∈ N , we have

(4.1) B 2 m = 2 2 m − 1 2 2 m − 1 − 1 ∑ k = 1 2 m ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j

and

(4.2) B 2 m = 2 2 m − 1 α ( 2 2 m − 1 − 1 ) + 2 2 m ( m − 1 ) + 1 ∑ k = 1 2 m ( k + α ) ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j ,

where α ≠ − 2 2 m ( m − 1 ) + 1 2 2 m − 1 − 1 is a real number and T ( 2 m + j , j ) is generated by (1.1) and given by (1.2).

Proof

In [14, Theorem 4.1], it was established that, when r < 0 is a real number, the series expansion

(4.3) sin x x r = 1 + ∑ m = 1 ∞ ( − 1 ) m ∑ k = 1 2 m ( − r ) k k ! ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j ( 2 x ) 2 m ( 2 m ) !

is convergent in x ∈ ( − π , π ) , where the rising factorial ( r ) k is defined by

( r ) k = ∏ ℓ = 0 k − 1 ( r + ℓ ) = r ( r + 1 ) ⋯ ( r + k − 1 ) , k ≥ 1 ; 1 , k = 0 .

Taking r = − 1 in (4.3) leads to

(4.4) x csc x = 1 + ∑ m = 1 ∞ ( − 1 ) m ∑ k = 1 2 m ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j ( 2 x ) 2 m ( 2 m ) !

for x ∈ ( − π , π ) . Comparing (2.3) with (4.4) and simplifying reveal (4.1).

Letting r = − 2 in (4.3) gives

(4.5) ( x csc x ) 2 = 1 + ∑ m = 1 ∞ ( − 1 ) m ∑ k = 1 2 m ( k + 1 ) ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j ( 2 x ) 2 m ( 2 m ) !

for x ∈ ( − π , π ) . Comparing (3.4) with (4.5) and making use of (4.1) yield

− 2 2 2 ! ∑ k = 1 2 ( k + 1 ) ∑ j = 1 k ( − 1 ) j k j R ( 2 + j , j , − j 2 ) 2 + j j = 2 B 2

and

(4.6) ∑ k = 1 m − 1 2 m 2 k ( 2 2 k − 1 − 1 ) ( 2 2 m − 2 k − 1 − 1 ) B 2 k B 2 m − 2 k = 2 2 m − 2 ∑ k = 1 2 m ( k − 1 ) ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j

for m ≥ 2 . Employing (3.1) in (4.6) results in

2 2 m − 1 2 2 m − 2 B 2 m + ∑ k = 1 m − 1 2 m 2 k B 2 k B 2 m − 2 k = ∑ k = 1 2 m ( k − 1 ) ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j

for m ≥ 2 . Further considering identity (3.5), which can be rewritten as

∑ k = 1 m − 1 2 m 2 k B 2 k B 2 m − 2 k = − ( 2 m + 1 ) B 2 m

for m ≥ 2 , we arrive at

2 2 m − 1 2 2 m − 2 B 2 m − ( 2 m + 1 ) B 2 m = ∑ k = 1 2 m ( k − 1 ) ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j ,

which can be reformulated as

(4.7) B 2 m = 2 2 m − 2 2 2 m − 2 ( 2 m − 3 ) + 1 ∑ k = 1 2 m ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j − ∑ k = 1 2 m k ∑ j = 1 k ( − 1 ) j k j T ( 2 m + j , j ) 2 m + j j .

Substituting (4.1) into (4.7) leads to

(4.8) B 2 m = 2 2 m − 1 2 2 m ( m − 1 ) + 1 ∑ k = 1 2 m k ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j , m ∈ N .

Finally, identity (4.8) can be reformulated as

B 2 m = 2 2 m − 1 2 2 m ( m − 1 ) + 1 ∑ k = 1 2 m ( k + α ) ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j − 2 2 m − 1 α 2 2 m ( m − 1 ) + 1 ∑ k = 1 2 m ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j = 2 2 m − 1 2 2 m ( m − 1 ) + 1 ∑ k = 1 2 m ( k + α ) ∑ j = 1 k ( − 1 ) j + 1 k j T ( 2 m + j , j ) 2 m + j j − 2 2 m − 1 α 2 2 m ( m − 1 ) + 1 2 2 m − 1 − 1 2 2 m − 1 B 2 m ,

where we used formula (4.1). Consequently, the explicit formula (4.2) is derived. Theorem 4.1 is thus proved.□

5 Simple proofs of series expansions of cosecant and cotangent

In this section, we supply simple proofs of series expansions (2.3), (2.4), (3.2), and (3.3) for (hyperbolic) cosecant and cotangent functions by the Euler formula in complex analysis.

At the site https://math.stackexchange.com/a/4427590/, there is a simple proof of the series expansion (2.3). We quote it with slight modifications as follows:

By the Euler formula

e i x = cos x + i sin x ,

we find the relation

sin x = e i x − e − i x 2 i .

Then,

csc x = 1 x 2 i x e i x − e − i x = 1 x ( 2 i x ) e ( 2 i x ) ∕ 2 e ( 2 i x ) − 1 = 1 x ∑ k = 0 ∞ B k 1 2 ( 2 i x ) k k ! = 1 x ∑ k = 0 ∞ ( 2 i ) k B k 1 2 x k k ! = 1 x ∑ k = 0 ∞ ( 2 i ) 2 k B 2 k 1 2 x 2 k ( 2 k ) ! = 1 x ∑ k = 0 ∞ ( − 1 ) k 2 2 k B 2 k 1 2 x 2 k ( 2 k ) ! = 1 x ∑ k = 0 ∞ ( − 1 ) k + 1 2 2 k 1 − 1 2 2 k − 1 B 2 k x 2 k ( 2 k ) ! = 2 x ∑ k = 0 ∞ ( − 1 ) k + 1 ( 2 2 k − 1 − 1 ) B 2 k x 2 k ( 2 k ) !

for ∣ x ∣ < π . The series expansion (2.3) is thus proved.

A similar and independent proof of the series expansion (2.3) is at the site https://math.stackexchange.com/a/649039/.

Since the relation cosh x = cos ( i x ) , the series expansion (2.4) can be derived from (2.3) readily.

As done in the above proof of the series expansion (2.3), we have

cot x = i ( e 2 i x + 1 ) e 2 i x − 1 = 1 2 x 2 i x e 2 i x e 2 i x − 1 + 2 i x e 2 i x − 1 = 1 2 x ∑ k = 0 ∞ [ B k ( 1 ) + B k ] ( 2 i x ) k k ! = 1 2 x ∑ k = 0 ∞ [ B 2 k ( 1 ) + B 2 k ] ( 2 i x ) 2 k ( 2 k ) ! = 1 x ∑ k = 0 ∞ ( − 1 ) k 2 2 k B 2 k x 2 k ( 2 k ) !

for ∣ x ∣ < π . The series expansion (3.2) is thus proved.

From the relation cot x = i coth ( i x ) or coth x = i cot ( i x ) , the series expansion (3.3) is derived immediately.

6 Conclusion

In this article, we presented two identities (2.1) and (3.1) in Theorems 2.1 and 3.1, in which the products B 2 j B 2 n − 2 j of the Bernoulli numbers B n are involved, provided two alternative proofs for these two identities (2.1) and (3.1), respectively, derived two closed-form formulas (4.1) and (4.2) in Theorem 4.1 for the Bernoulli numbers B 2 m in terms of central factorial numbers of the second kind T ( n , k ) , and supplied simple proofs of series expansions of the (hyperbolic) cosecant and cotangent functions in Section 5.

# Dedicated to Professor Ravi Prakash Agarwal at Department of Mathematics, Texas A&M University-Kingsville, USA.

Acknowledgment

The authors are thankful to anonymous referees for their careful corrections to and helpful comments on the original version of this article.

  1. Funding information: The first two authors, X.-Y. Chen and L. Wu, were supported by the College Scientific Research Project of Inner Mongolia (Grant No. NJZZ19144 and Grant No. NJZY19156), the Development Plan for Young Technological Talents in Colleges and Universities of Inner Mongolia (Grant No. NJYT22051), the Natural Science Foundation Project of Inner Mongolia (Grant No. 2021LHMS05030), the Basic Scientific Research Business Expense Project of Colleges and Universities Directly Under Inner Mongolia Autonomous Region (Grant No. GXKY22045), the Intelligent Agricultural Machinery Equipment and Technology Team of Inner Mongolia Minzu University, the Teaching Team of “Fundamentals of Control Engineering”, and the National Natural Science Foundation of China (Grant No. 61440041). The third and corresponding author, D. Lim, was supported by the National Research Foundation of Korea under Grant NRF-2021R1C1C1010902, Republic of Korea.

  2. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare that they have no conflict of competing interests.

  4. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

  5. Ethical approval: The conducted research is not related to either human or animal use.

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Received: 2022-04-11
Revised: 2022-07-29
Accepted: 2022-09-12
Published Online: 2022-11-16

© 2022 Xue-Yan Chen et al., published by De Gruyter

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https://doi.org/10.1515/dema-2022-0166
Keywords for this article
Bernoulli number; identity; product; cosecant; cotangent; hyperbolic cosecant; hyperbolic cotangent; series expansion; central factorial number of the second kind; closed-form formula
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