On the twisted Daehee polynomials with q-parameter
© Park; licensee Springer. 2014
Received: 16 September 2014
Accepted: 17 November 2014
Published: 2 December 2014
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.
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 .
Note that for each .
When , are called the Bernoulli numbers of order r, and in the special case, , are called the ordinary Bernoulli polynomials.
where is the cyclic group of order .
When , are called the Daehee numbers with q-parameter.
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
By (2.2) and (2.3), we obtain the following theorem.
By (2.4) and (2.5), we obtain the following corollary.
By (1.6), (2.7), and (2.8), we obtain the following corollary.
where n is a nonnegative integer and . In the special case, , are called the Daehee numbers of order k with q-parameter.
By (2.13), (2.14), and (2.15), we obtain the following theorem.
In the special case , are called the twisted Daehee numbers of the second kind with q-parameter.
By (1.10), it is easy to show that . Thus, from (2.19), we have the following theorem.
By (2.20) and (2.21), we obtain the following theorem.
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 (1.10), we know that . Hence, by (2.24), we obtain the following theorem.
By (2.25) and (2.26), we obtain the following theorem.
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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