Abstract
As is well known, power sums of consecutive nonnegative integers can be expressed in terms of Bernoulli polynomials. Also, it is well known that alternating power sums of consecutive nonnegative integers can be represented by Euler polynomials. In this paper, we show that power sums of consecutive positive odd q-integers can be expressed by means of type 2 q-Bernoulli polynomials. Also, we show that alternating power sums of consecutive positive odd q-integers can be represented by virtue of type 2 q-Euler polynomials. The type 2 q-Bernoulli polynomials and type 2 q-Euler polynomials are introduced respectively as the bosonic p-adic q-integrals on \(\mathbb{Z}_{p}\) and the fermionic p-adic q-integrals on \(\mathbb{Z}_{p}\). Along the way, we will obtain Witt type formulas and explicit expressions for those two newly introduced polynomials.
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1 Introduction
Let p be a fixed odd prime number. Throughout this paper, \(\mathbb{Z}_{p}\), \(\mathbb{Q}_{p}\), and \(\mathbb{C}_{p}\) will denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of \(\mathbb{Q}_{p}\), respectively. Let \(|\cdot |_{p}\) be the p-adic norm which is normalized as \(|p|_{p}=\frac{1}{p}\). Let q be an indeterminate such that \(q \in {\mathbb{C}}_{p}\) with \(|1-q|_{p} < p^{-\frac{1}{p-1}}\), and \([x]_{q} = \frac{1-q^{x}}{1-q}\). Note that \([x]_{-q} = \frac{1-(-q)^{x}}{1+q}\).
The bosonic p-adic q-integrals on \(\mathbb{Z}_{p}\) are defined by Kim as
where f is a uniformly differentiable function on \(\mathbb{Z}_{p}\).
In [1, 2], Carlitz considered the q-Bernoulli numbers which are given by the recurrence relation:
with the usual convention about replacing \(\beta _{q}^{n}\) by \(\beta _{n,q}\).
In [6, 7], Kim gave the following Witt type formula:
Carlitz also defined the q-Bernoulli polynomials by
where n is a nonnegative integer.
An integral representation for \(\beta _{n,q}(x)\), \((n \geq 0)\) was given by Kim as follows:
In [6, 8, 10], Kim introduced the modified q-Bernoulli polynomials as the p-adic q-integral on \(\mathbb{Z}_{p}\) given by
For \(x=0\), \(B_{n,q} = B_{n,q}(0)\) are called the modified q-Bernoulli numbers.
From (1.1), we note that
with the usual convention about replacing \(B_{q}^{n}\) by \(B_{n,q}\).
By (1.6), we easily get
It is well known that
Here \(B_{k}(x)\) are the Bernoulli polynomials given by
and \(B_{n} = B_{n}(0)\) are called the Bernoulli numbers.
In [8], Kim proved that the power sums of consecutive nonnegative q-integers are given by
Now, we consider the power sums of consecutive odd positive q-integers and ask the following question:
In addition, we ask the following question:
We will see that (1.10) can be expressed in terms of type 2 q-Bernoulli polynomials and (1.11) by virtue of type 2 q-Euler polynomials. Here we note that the type 2 q-Bernoulli polynomials are represented by bosonic p-adic q-integrals on \(\mathbb{Z}_{p}\) and the type 2 q-Euler polynomials by fermionic p-adic q-integrals on \(\mathbb{Z}_{p}\).
2 Type 2 q-Bernoulli polynomials and numbers
From (1.1), we have
By using (2.1) and induction, we get
where n is a positive integer.
In view of (1.6), we consider the generating function of the type 2 q-Bernoulli polynomials given by the following p-adic q-integral on \(\mathbb{Z}_{p}\):
From (2.3), we have
For \(x=0\), \(b_{n,q} = b_{n,q}(0)\) are called the type 2 q-Bernoulli numbers.
By (2.4), we get
By (2.5), we easily get
Theorem 2.1
For \(n \geq 0\), we have
By (2.7), we can derive the generating function for the type 2 q-Bernoulli numbers as follows:
From (2.4), we note that
and that
From (2.4), we easily get
From (2.2), we note that
Therefore, by (2.12), we obtain the following theorem.
Theorem 2.2
For \(m \geq 0\) and \(n \in \mathbb{N}\), we have
From (2.13), we note that
By (2.14), we get the following corollary.
Corollary 2.3
For \(n \in \mathbb{N}\) and \(m \geq 0\), we have
Example
Here we check formula (2.15) for \(m=1\). First, we observe that
By (2.16), we get
We now show that (2.17) agrees with the result in (2.15). For this, we first note the following from (2.7):
Then, from (2.15), we have
3 Type 2 q-Euler polynomials and numbers
It is known that the fermionic p-adic q-integrals on \(\mathbb{Z} _{p}\) are defined by Kim as
where \([x]_{-q} = \frac{1-(-q)^{x}}{1+q}\).
From (3.1), we note that
By (3.2), we get
and
where \(f_{n}(x) = f(x+n)\), with \(n \in \mathbb{N}\).
As is known, Carlitz considered q-Euler numbers given by the recurrence relation
with the usual convention about replacing \(E_{q}^{l}\) by \(E_{l,q}\) (see [1, 2]).
In [11], Kim obtained the Witt type formula for Carlitz’s q-Euler numbers which is represented by the fermionic p-adic q-integrals on \(\mathbb{Z}_{p}\)
From (3.6), we note that
By (3.7), we readily see that the generating function for Carlitz’s q-Euler numbers is given by
It is known that
where \(n \in \mathbb{N}\) with \(n \equiv 1(\mathrm{mod}~2)\). Note that equation (3.9) is an alternating sum of powers of consecutive positive q-integers.
Now, we consider an alternating sum of powers of consecutive positive odd q-integers which are given by
Let us define the type 2 q-Euler polynomials which are given by
When \(x=0\), \(\mathcal{E}_{n,q} = \mathcal{E}_{n,q}(0)\), \((n \geq 0)\) are called the type 2 q-Euler numbers.
From (3.11), we note that
By (3.12), we get the following generating function for the q-Euler polynomials:
Theorem 3.1
For \(m \geq 0\), we have
From (3.11), we have
Also, by (3.3), we get
Therefore, by (3.15) and (3.16), we obtain the following theorem.
Theorem 3.2
For \(m \geq 0\), we have
In particular,
Let n be a positive integer with \(n \equiv 1(\mathrm{mod}~2)\). From (3.4), we have
Therefore, by (3.20), we obtain the following theorem.
Theorem 3.3
For \(n \in \mathbb{N}\) with \(n \equiv 1(\mathrm{mod}~2)\) and \(m \geq 0\), we have
4 Conclusions
In an introductory calculus class, the following formulas are proved by mathematical induction and used in Riemann sum evaluations of some definite integrals:
The problem of finding formulas for power sums of consecutive nonnegative integers has captivated mathematicians for many centuries. Even since generalized formulas for the power sums, \(S_{k} (n) = \sum_{l=0}^{n} l^{k}\), were established, the various representations and number-theoretic properties have been studied by Faulhaber. In this paper, we studied the q-analogues of Faulhaber’s well-known formula expressing the power sums in terms of Bernoulli polynomials. Indeed, we showed that power sums of consecutive positive odd q-integers can be expressed by means of type 2 q-Bernoulli polynomials. Also, we showed that alternating power sums of consecutive positive odd q-integers can be represented by virtue of type 2 q-Euler polynomials. The type 2 q-Bernoulli polynomials and type 2 q-Euler polynomials were introduced respectively as the bosonic p-adic q-integrals on \(\mathbb{Z}_{p}\) and the fermionic p-adic q-integrals on \(\mathbb{Z}_{p}\). Along the way, we also obtained Witt type formulas and explicit expressions for those two newly introduced polynomials.
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Funding
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).
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Kim, D.S., Kim, T., Kim, H.Y. et al. A note on type 2 q-Bernoulli and type 2 q-Euler polynomials. J Inequal Appl 2019, 181 (2019). https://doi.org/10.1186/s13660-019-2131-6
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DOI: https://doi.org/10.1186/s13660-019-2131-6