Abstract
We present several formulas involving the classical Bernoulli numbers and polynomials. Among others, we extend an identity for Bernoulli polynomials published by Wu et al. (Fibonacci Quart 42:295-299, 2004).
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1 Introduction and Statement of the Results
In 2001, Momiyama [7] used methods from p-adic analysis to prove the following remarkable identity for the classical Bernoulli numbers \(B_{\nu }\), defined by
Proposition 1
For all nonnegative integers m and n with \(m+n>0\), we have
From (1.1) with \(m=n\) we obtain
which is attributed to von Ettingshausen [8, pp. 284-285]; see also Kaneko [5]. In 2004, Wu et al. [9] used properties of power series to prove an interesting generalization of (1.1) involving the Bernoulli polynomials
Proposition 2
For all nonnegative integers m and n and complex numbers z, we have
If \(m+n>0\) and \(z=0\), then (1.2) reduces to (1.1). Chen and Sun [2] applied Zeilberger’s algorithm to prove (1.2). We show that (1.2) can be written in a slightly shorter and more elegant form as follows.
Theorem 1
Let m and n be nonnegative integers. Then, for \(z\in {\mathbb {C}}\),
We apply some basic properties of Bernoulli polynomials to settle (1.3). In particular, we provide a new proof of (1.2).
The main aim of this paper is to present a new generalization of (1.2). We define
The following reciprocity formula holds. (As usual, if \(p<0\), then \(B_p(z)=0\).)
Theorem 2
Let m, n and k be nonnegative integers. Then, for \(z\in {\mathbb {C}}\),
Remark 1
The special case \(k=0\) gives (1.2).
If we set \(z=0\), then (1.4) yields an extension of Momiyama’s identity (1.1).
Corollary 1
Let m, n, k be nonnegative integers. Then
where
An application of Theorem 2 leads to an identity for the alternating sum
We obtain the following companion to (1.5).
Corollary 2
Let k, m, n be nonnegative integers with \(k+2\le \min (m,n)\). Then
A second application of Theorem 2 gives a reciprocity formula for the polynomial
Corollary 3
Let k, m, n be nonnegative integers. Then, for \(z\in {\mathbb {C}}\),
Remark 2
From (1.7) with \(z=-1/2\) we get
By using differentiation or integration certain summation formulas involving Bernoulli polynomials lead to interesting new identities. Hereby, the recurrence relation \(B'_n(x)=nB_{n-1}(x)\) plays an important role. We show that by applying an integral formula and (1.3) we obtain the following counterpart of
which is due to Gessel [3].
Corollary 4
Let m and n be nonnegative integers. Then
Remark 3
Combining (1.8) and (1.9) gives
2 Proofs
I. We need the following formulas:
These formulas can be found in Abramowitz and Stegun [1, (23.1.7), (23.1.8), (23.1.9)]. The integral formula
is given in Moll and Vignat [6]. Applying (2.2) and (2.3) leads to
The next two identities can be found in Gould [4, (1.13), (3.49)]:
Applying the binomial theorem gives
II. We show that the identities (1.2) and (1.3) are equivalent. Let
with
Using (2.2) gives
This implies that (1.2) and (1.3) are equivalent.
III. Now, we prove the theorems and corollaries stated in Sect. 1.
Proof of Theorem 1
Using (2.8), (2.2), (2.1) with \(x=-z\), \(y=1\) and the elementary formula
with \(r=\nu + n\) gives
Applying (2.5) we conclude that the inner sum is equal to zero, if \(0\le j\le m-1\). Thus
Using (2.6) with \(k=m+1\), \(r=j+m+1\), \(x=m+n+2\) we get
From (2.9) and (2.10) we obtain
This settles (1.3). \(\square \)
Proof of Theorem 2
Using (2.1) gives
Next, we exchange m and n and we replace x and z by \(-x\) and \(-z\), respectively. Then we obtain from (2.11):
Moreover, we have
We apply (1.2) with \(x+z\) instead of z and (2.11), (2.12), (2.13). A comparison of the coefficients yields
as claimed. \(\square \)
Proof of Corollary 2
We define
and
Then
Using
gives
and
We apply Theorem 2 with \(z=1\) and (2.14), (2.15). This yields
Next, we exchange m and n, multiply by \((-1)^{k+1}\), and use (1.5). Then
From (2.16) and (2.17) we obtain
with
If \(0\le p\le m\), then we conclude from (2.5) that
Since \(k+2\le \min (m,n)\), we obtain from (2.19) that \(F(m,n;k)=0\), so that (2.18) implies (1.6). \(\square \)
Proof of Corollary 3
We present two proofs. First proof. A short calculation gives that (1.7) holds for \(k=0\) and \(k=1\). Moreover, since
we conclude that (1.7) is valid for \(m=0\) and \(n=0\). Let \(m\ge 1\), \(n\ge 1\), \(k\ge 2\). Using (2.2) gives
and
From Theorem 2 we obtain
Combining (2.20), (2.21) and (2.22) leads to (1.7).\(\square \)
Second proof. We have
and
\(\square \)
Proof of Corollary 4
We assume that \(m>n\ge 0\). Since, for \(m\ge 1\),
we conclude that (1.9) holds for \(n=0\). Next, let \(m>n\ge 1\). Using (2.3) gives
and from (2.4) and (2.7) with \(r=n\), \(s=m\), \(t=-1/2\) we get
Applying (1.3), (2.23), (2.24) and
we obtain (1.9). \(\square \)
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References
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Abel, U., Alzer, H. Formulas for Bernoulli Numbers and Polynomials. Results Math 79, 246 (2024). https://doi.org/10.1007/s00025-024-02273-6
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DOI: https://doi.org/10.1007/s00025-024-02273-6