Abstract
In this paper, we investigate some identities of q-extensions of special polynomials which are derived from the fermonic q-integral on and the bosonic q-integral on .
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1 Introduction
Let p be a fixed odd prime number. Throughout this paper, , , and will denote the ring of p-adic integers, the field of p-adic rational numbers and the completion of algebraic closure of , respectively. Let q be an indeterminate in with and be the space of all uniformly differentiable functions on . The q-analog of x is defined as . Note that . For , the bosonic p-adic q-integral on is defined by Kim to be
and the fermionic p-adic q-integral on is also defined by Kim to be
From (1.1) and (1.2), we have
and
As is well known, the q-analog of the Bernoulli polynomials is given by the generating function to be
and the q-analog of the Euler polynomials is given by
The higher-order q-Daehee polynomials are given by
where with .
Now, we define the q-analog of the Changhee polynomials, which are given by the generating function to be
In this paper, we investigate some properties for the q-analog of several special polynomials which are derived from the bosonic or fermionic p-adic q-integral on .
2 Some special q-polynomials
In this section, we assume that with . Now, we define the higher-order q-Bernoulli numbers,
When , are called the higher-order q-Bernoulli numbers.
We also consider the higher-order q-Daehee polynomials as follows:
When , are called the higher-order q-Daehee numbers.
From (1.3), we can derive the following equation:
Thus, by (2.3), we get
By replacing t by in (2.2), we get
and
Thus, by (2.5) and (2.6), we get
Therefore, by (2.4) and (2.7), we obtain the following theorem.
Theorem 1 For , we have
and
where is the Stirling number of the second kind.
From (2.1), by replacing t by , we obtain
where is the Stirling number of the first kind.
Therefore, by (2.2) and (2.8), we obtain the following theorem.
Theorem 2 For , we have
Now, we define the higher-order q-Changhee polynomials as follows:
When , are called the higher-order q-Changhee numbers.
From (1.4), we note that
Thus, by (2.10), we get
In view of (1.6), we define the higher-order q-Euler polynomials which are given by the generating function to be
From (2.10), we note that
Therefore, by (2.11) and (2.13), we obtain the following theorem.
Theorem 3 For , we have
By replacing t by in (2.9), we get
and
Therefore, by (2.12), (2.14), and (2.15), we obtain the following theorem.
Theorem 4 For , we have
Now, we consider the q-analog of the higher-order Cauchy polynomials, which are defined by the generating function to be
When , are called the higher-order q-Cauchy numbers. Indeed,
where are called the higher-order Cauchy polynomials.
We observe that
and
By (2.18) and (2.19), we get
Therefore, by (2.20), we obtain the following theorem.
Theorem 5 For , we have
For , we define the q-analog of the Bernoulli-Euler mixed-type polynomials of order as follows:
Then, by (2.21), we get
It is easy to show that
Therefore, by (2.22) and (2.23), we obtain the following theorem.
Theorem 6 For , we have
By replacing t by in (2.22), we get
and
Therefore, by (2.24) and (2.25), we obtain the following theorem.
Theorem 7 For , we have
Let us consider the q-analog of the Daehee-Changhee mixed-type polynomials of order as follows: for ,
Thus, by (2.26), we get
and
Now, we observe that
Therefore, by (2.27), (2.28), and (2.29), we obtain the following theorem.
Theorem 8 For , we have
and
Now, we consider the q-extension of the Cauchy-Changhee mixed-type polynomials of order as follows: for ,
Thus, by (2.30), we get
Note that
Therefore, by (2.31), (2.32), and (2.33), we obtain the following theorem.
Theorem 9 For , we have
and
Finally, we define the q-extension of the Cauchy-Daehee mixed-type polynomials of order as follows:
Thus, by (2.34), we get
Therefore, by (2.35), we obtain the following equation:
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This paper is supported by grant No. 14-11-00022 of Russian Scientific fund.
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Dolgy, D.V., Kim, D.S., Kim, T. et al. Some identities of special q-polynomials. J Inequal Appl 2014, 438 (2014). https://doi.org/10.1186/1029-242X-2014-438
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DOI: https://doi.org/10.1186/1029-242X-2014-438