Abstract
The degenerate Bernoulli polynomials were introduced by Carlitz and rediscovered later by Ustiniv under the name of Korobov polynomials of the second kind (see Carlitz in Arch. Math. (Basel) 7:28-33, 1956; Util. Math. 15:51-88, 1979). In this paper, we study q-analogs of degenerate Bernoulli polynomials and give some formulas related to these polynomials.
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1 Introduction
Let p be a fixed 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 \(\mathbb{Q}_{p}\). The p-adic norm is normalized as \(\vert p\vert _{p}=\frac{1}{p}\). Let \(UD (\mathbb{Z}_{p})\) be the space of all \(\mathbb{C}_{p}\)-valued uniformly differentiable functions on \(\mathbb{Z}_{p}\), and let q be an indeterminate such that \(\vert 1-q\vert _{p}< p^{-\frac{1}{p-1}}\). The q-extension of the number x is defined as \([x ]_{q}=\frac{1-q^{x}}{1-q}\). Note that \(\lim_{q\rightarrow1} [x ]_{q}=x\). For \(f\in UD (\mathbb{Z}_{p})\), the p-adic q-integral on \(\mathbb{Z}_{p}\) is defined by Kim to be
The ordinary p-adic invariant integral on \(\mathbb{Z}_{p}\) is given by
From (1.1), we can derive the following integral equation:
where \(f_{1} (x )=f (x+1 )\).
For \(\lambda,t\in\mathbb{C}\) with \(\vert \lambda t\vert _{p}< p^{-\frac {1}{p-1}}\), the degenerate Bernoulli polynomials are defined as
When \(x=0\), \(\beta_{n} (\lambda )=\beta_{n} (0\mid \lambda )\) are called the degenerate Bernoulli numbers. As is well known, the Bernoulli polynomials of the second kind are defined by the generating function:
When \(x=0\), \(b_{n}=b_{n} (0 )\) are called the Bernoulli numbers of the second kind. The Daehee polynomials are also given by the generating function:
Now, we define the q-analogs of Bernoulli polynomials of the second kind as follows:
Note that \(\lim_{q\rightarrow1}b_{n,q} (x )=b_{n} (x )\).
The q-analogs of Daehee polynomials are defined by the generating function to be
When \(x=0\), \(b_{n,q}=b_{n,q} (0 )\) are called the q-analogs of Bernoulli numbers of the second kind and \(D_{n,q}=D_{n,q} (0 )\) are called the q-analogs of Daehee numbers.
where \((x )_{n}=x (x-1 )\cdots (x-n+1 )=\sum_{l=0}^{n}S_{1} (n,l )x^{l}\).
In this paper, we study q-analogs of degenerate Bernoulli polynomials and give some formulas related to these polynomials.
2 q-Analogs of degenerate Bernoulli polynomials
In this section, we assume that \(\lambda,t\in\mathbb{C}_{p}\) with \(\vert \lambda t\vert < p^{-\frac{1}{p-1}}\). Let us take \(f (y )= (1+\lambda t )^{\frac{y}{\lambda}}\). Then by (1.3), we get
where \(D_{n,q} (x\mid\lambda )\) are called the q-analogs of λ-Daehee polynomials. When \(x=0\), \(D_{n,q} (\lambda )=D_{n,q} (0\mid\lambda )\) are called the q-analogs of λ-Daehee numbers.
From (1.3), we can easily derive the following equation:
where \(f_{n} (x )=f (x+n )\).
Thus, by (2.2), we get
By (2.3), we get
It is easy to see that
where \((x\mid\lambda )_{l}=x (x-\lambda )\cdots (x- (l-1 )\lambda )\) (see [1–17]).
Now, we define the q-analogs of degenerate Bernoulli polynomials as follows:
When \(x=0\), \(\beta_{n,q} (\lambda )=\beta_{n,q} (0\mid \lambda )\) are called the q-analogs of degenerate Bernoulli numbers.
Therefore, by (2.6), we obtain the following theorem.
Theorem 2.1
For \(m\in\mathbb{N}\), we have
We observe that
Now, we define the q-analogs of degenerate Bernoulli polynomials of the second kind as follows:
When \(x=0\), \(b_{n} (\lambda )=b_{n} (0\mid\lambda )\) are called the q-analogs of degenerate Bernoulli numbers of the second kind.
Indeed, we note that \(\lim_{\lambda\rightarrow1}b_{n,q} (x\mid \lambda )=b_{n,q} (x )\).
By (2.1), we easily get
From (2.1) and (2.8), we note that
Therefore, by (2.7) and (2.10), we obtain the following theorem.
Theorem 2.2
For \(n\ge0\), we have
As is well known, the Apostol-Bernoulli polynomials are defined by the generating function:
By (2.5) and (2.11), we get \(\lim_{\lambda \rightarrow 0}\beta_{n,q} (x\mid\lambda )=B_{n} (x\mid q )\) (\(n\ge0\)).
From (2.8), we can derive the following equation:
By replacing t by \(\frac{1}{\lambda} (e^{\lambda t}-1 )\) in (2.7), we get
where \(B_{n,q} (x )\) are the q-Bernoulli polynomials which are given by the generating function:
On the other hand,
Therefore, by (2.13) and (2.15), we obtain the following theorem.
Theorem 2.3
For \(n\ge0\), we have
From (2.7), we have
Note that
and
By (2.16), (2.17), and (2.18), we get
Therefore, by (2.19), we obtain the following theorem.
Theorem 2.4
For \(n\ge0\), we have
For \(k\in\mathbb{N}\), we define the q-analogs of degenerate Bernoulli polynomials of order k as follows:
When \(x=0\), \(\beta_{n,q} (\lambda )=\beta_{n,q} (0\mid \lambda )\) are called the q-analogs of degenerate Bernoulli numbers of order k.
From (2.20), we note that
where \(B_{n}^{ (k )} (x\mid q )\) are called the higher-order Apostol-Bernoulli polynomials.
Thus, by (2.21), we get \(\lim_{\lambda\rightarrow0}\beta _{n,q}^{ (k )} (x\mid\lambda )=B_{n}^{ (k )} (x\mid q )\) (\(n\geq0 \)).
For \(k\in\mathbb{N}\), by (2.20), we get
Now, we define the q-analogs of higher-order degenerate Bernoulli polynomials of the second kind as follows:
When \(x=0\), \(b_{n,q}^{ (k )} (\lambda )=b_{n,q}^{ (k )} (0\mid\lambda )\) are called the q-analogs of higher-order degenerate Bernoulli numbers of the second kind. Note that \(\lim_{\lambda\rightarrow 1}b_{n,q}^{ (k )} (x\mid\lambda )=b_{n,q}^{ (k )} (x )\), and \(\lim_{q\rightarrow1}b_{n,q}^{ (k )} (x )=b_{n}^{ (k )} (x )\).
From (2.23), we can derive the following equation:
It is easy to show that
where \(B_{n,q}^{ (k )} (x )\) are called the q-Bernoulli polynomials of order k.
Thus, by (2.25), we get
Therefore, by (2.22), (2.24), and (2.26), we obtain the following theorem.
Theorem 2.5
For \(n\ge0\), we have
Remark
We define the q-analogs of λ-Daehee polynomials of order k as follows:
From (2.27), we have
Thus, by (2.28), we get
From (2.22), (2.24), and (2.29), we have
From (2.1), we can derive the following equation:
Note that \(D_{n} (1-x\mid\lambda )= (-1 )^{n}D_{n} (x\mid-\lambda ) \) (\(n\ge0\)).
By (2.31), we get
where \(\langle x\vert \lambda \rangle _{n}=x (x+\lambda )\cdots (x+ (n-1 )\lambda )\).
Note that
From (2.32) and (2.33), we have
By (2.7), we get
By (2.8), we get
Therefore, by (2.32), (2.33), (2.35), and (2.36), we get
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Kim, D.S., Kim, T. & Dolgy, D.V. On q-analogs of degenerate Bernoulli polynomials. Adv Differ Equ 2015, 194 (2015). https://doi.org/10.1186/s13662-015-0522-7
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DOI: https://doi.org/10.1186/s13662-015-0522-7