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On the twisted Daehee polynomials with q-parameter
Advances in Difference Equations volume 2014, Article number: 304 (2014)
Abstract
The n th twisted Daehee numbers with q-parameter are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, and we derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.
1 Introduction
Let p be a fixed prime number. Throughout this paper, , , and will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completions of algebraic closure of . The p-adic norm is defined .
When one talks of q-extension, q is variously considered as an indeterminate, a complex , or a p-adic number . If , one normally assumes that . If , then we assume that so that for each . Throughout this paper, we use the notation
Note that for each .
Let be the space of uniformly differentiable functions on . For , the p-adic invariant integral on is defined by Kim as follows:
Let be the translation of f with . Then, by (1.1), we get
As is well known, the Stirling number of the first kind is defined by
and the Stirling number of the second kind is given by the generating function to be
Unsigned Stirling numbers of the first kind are given by
Note that if we replace x to −x in (1.3), then
Hence .
For , the Bernoulli polynomials of order r are defined by the generating function to be
When , are called the Bernoulli numbers of order r, and in the special case, , are called the ordinary Bernoulli polynomials.
For , let be the p-adic locally constant space defined by
where is the cyclic group of order .
We assume that q is an indeterminate in with . Then we define the q-analog of a falling factorial sequence as follows:
Note that
Recently, DS Kim and T Kim introduced the Daehee polynomials as follows:
When , are called the nth Daehee numbers. From (1.8), we can derive the generating function to be
In addition, DS Kim et al. consider the Daehee polynomials with q-parameter, which are defined by the generating function to be
When , are called the Daehee numbers with q-parameter.
From the viewpoint of a generalization of the Daehee polynomials with q-parameter, we consider the twisted Daehee polynomials with q-parameter, defined to be
where with and .
In this paper, we give a p-adic integral representation of the twisted Daehee polynomials with q-parameter, which is called the Witt-type formula for the twisted Daehee polynomials with q-parameter. We can derive some interesting properties related to the n th twisted Daehee polynomials with q-parameter.
2 Witt-type formula for the n th twisted Daehee polynomials with q-parameter
First, we consider the following integral representation associated with falling factorial sequences:
By (2.1),
where with . For with , put . By (1.1), we get
By (2.2) and (2.3), we obtain the following theorem.
Theorem 2.1 For , we have
In (2.3), by replacing t by , we have
and
By (2.4) and (2.5), we obtain the following corollary.
Corollary 2.2 For , we have
By Theorem 2.1,
By (1.2), we can derive easily that
and so
By (1.6), (2.7), and (2.8), we obtain the following corollary.
Corollary 2.3 For , we have
From now on, we consider twisted Daehee polynomials of order with q-parameter. Twisted Daehee polynomials of order with q-parameter are defined by the multivariant p-adic invariant integral on :
where n is a nonnegative integer and . In the special case, , are called the Daehee numbers of order k with q-parameter.
From (2.9), we can derive the generating function of as follows:
Note that, by (2.9),
Since
we can derive easily
Thus, by (2.11) and (2.12), we have
In (2.10), by replacing t by , we get
and
By (2.13), (2.14), and (2.15), we obtain the following theorem.
Theorem 2.4 For and , we have
and
Now, we consider the twisted Daehee polynomials of the second kind with q-parameter as follows:
In the special case , are called the twisted Daehee numbers of the second kind with q-parameter.
By (2.16), we have
and so we can derive the generating function of by (1.1) as follows:
From (1.3), (1.6), and (2.17), we get
By (1.10), it is easy to show that . Thus, from (2.19), we have the following theorem.
Theorem 2.5 For , we have
By replacing t by in (2.18), we have
and
By (2.20) and (2.21), we obtain the following theorem.
Theorem 2.6 For , we have
Now, we consider higher-order twisted Daehee polynomials of the second kind with q-parameter. Higher-order twisted Daehee polynomials of the second kind with q-parameter are defined by the multivariant p-adic invariant integral on :
where n is a nonnegative integer and . In the special case, , are called the higher-order twisted Daehee numbers of the second kind with q-parameter.
From (2.22), we can derive the generating function of as follows:
By (2.22),
From (1.10), we know that . Hence, by (2.24), we obtain the following theorem.
Theorem 2.7 For , we have
In (2.23), by replacing t by , we get
and
By (2.25) and (2.26), we obtain the following theorem.
Theorem 2.8 For and , we have
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Acknowledgements
The author is grateful for the valuable comments and suggestions of the referees. This paper was supported by the Sehan University Research Fund in 2014.
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Park, JW. On the twisted Daehee polynomials with q-parameter. Adv Differ Equ 2014, 304 (2014). https://doi.org/10.1186/1687-1847-2014-304
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DOI: https://doi.org/10.1186/1687-1847-2014-304
Keywords
- Bernoulli polynomials
- Daehee polynomials with q-parameter
- p-adic invariant integral