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.
1 Introduction
The Bernoulli numbers
More generally, the Bernoulli polynomials
for
The central factorial numbers of the second kind
In [7, Chapter 6, equation (26)], it was established that
with
In this article, we present two identities involving the products
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
Theorem 2.1
For
For
First proof
On page 42 in the handbook [11], the series expansions
and
for
for
Since the function
Making use of identity (2.1) and the fact that
listed in [12, p. 805, 23.1.15], we finally deduce
for
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
The left-hand side of identity (2.1) can be broken into four parts as
The first part is
at
3 The second identity and its two proofs
The second identity involving the products
Theorem 3.1
For
First proof
It is known that
Hence, making use of the series expansions (2.3) and (2.4) and considering the series expansions
and
for
and
for
or
and comparing coefficients of the terms
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
which can be proved using the definition of
See also [13, p. 595, 24.14.2]. Especially when
Since
Hence, we acquire
If
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
Theorem 4.1
For
and
where
Proof
In [14, Theorem 4.1], it was established that, when
is convergent in
Taking
for
Letting
for
and
for
for
for
which can be reformulated as
Substituting (4.1) into (4.7) leads to
Finally, identity (4.8) can be reformulated as
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
we find the relation
Then,
for
A similar and independent proof of the series expansion (2.3) is at the site https://math.stackexchange.com/a/649039/.
Since the relation
As done in the above proof of the series expansion (2.3), we have
for
From the relation
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
Acknowledgment
The authors are thankful to anonymous referees for their careful corrections to and helpful comments on the original version of this article.
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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.
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Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors declare that they have no conflict of competing interests.
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Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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Ethical approval: The conducted research is not related to either human or animal use.
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