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Some identities of q-Euler polynomials arising from q-umbral calculus
Journal of Inequalities and Applications volume 2014, Article number: 1 (2014)
Recently, Araci-Acikgoz-Sen derived some interesting identities on weighted q-Euler polynomials and higher-order q-Euler polynomials from the applications of umbral calculus (see (Araci et al. in J. Number Theory 133(10):3348-3361, 2013)). In this paper, we develop the new method of q-umbral calculus due to Roman, and we study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give some interesting identities on our q-Euler polynomials related to the q-Bernoulli numbers and polynomials of Hegazi and Mansour.
Throughout this paper we will assume q to be a fixed real number between 0 and 1. We define the q-shifted factorials by
If x is a classical object, such as a complex number, its q-version is defined as . We now introduce the q-extension of exponential function as follows:
where with .
The Jackson definite q-integral of the function f is defined by
The q-difference operator is defined by
By using an exponential function , Hegazi and Mansour defined q-Bernoulli polynomials as follows:
In the special case, , are called the n th q-Bernoulli numbers.
From (1.5), we can easily derive the following equation:
In the next section, we will consider new q-extensions of Euler numbers and polynomials by using the method of Hegazi and Mansour. More than five decades ago, Carlitz  defined a q-extension of Euler polynomials. In a recent paper (see ), Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials which differ from Carlitz’s q-Bernoulli numbers and polynomials. By using the method of Kupershmidt, Hegazi and Mansour also introduced a new q-extension of Bernoulli numbers and polynomials (see [1, 3, 4]). From the q-exponential function, Kurt and Cenkci derived some interesting new formulae of q-extension of Genocchi polynomials. Recently, several authors have studied various q-extensions of Bernoulli and Euler polynomials (see [1–6, 8–11]). Let ℂ be the complex number field, and let ℱ be the set of all formal power series in variable t over ℂ with
Let and let be the vector space of all linear functionals on ℙ. denotes the action of linear functional L on the polynomial , and it is well known that the vector space operations on are defined by
For , we define the linear functional on ℙ by setting
From (1.7) and (1.8), we note that
where is the Kronecker symbol.
Let us assume that . Then by (1.9) we easily see that . That is, . Additionally, the map is a vector space isomorphism from onto ℱ. Henceforth ℱ denotes both the algebra of formal power series in t and the vector space of all linear functionals on ℙ, and so an element of ℱ will be thought of as a formal power series and a linear functional. We call it the q-umbral algebra. The q-umbral calculus is the study of q-umbral algebra. By (1.2) and (1.3), we easily see that and so for . The order of the power series is the smallest integer for which does not vanish. If , then is called an invertible series. If , then is called a delta series (see [7, 9, 11, 12]). For , we have . Let and . Then we have
From (1.10), we have
By (1.11), we get
Thus from (1.12), we note that
Let with and . Then there exists a unique sequence () of polynomials such that (). The sequence is called the q-Sheffer sequence for which is denoted by . Let . For and , we have
Recently, Araci-Acikgoz-Sen derived some new interesting properties on the new family of q-Euler numbers and polynomials from some applications of umbral algebra (see ). The properties of q-Euler and q-Bernoulli polynomials seem to be of interest and worthwhile in the areas of both number theory and mathematical physics. In this paper, we develop the new method of q-umbral calculus due to Roman and study a new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give new explicit formulas on q-Euler polynomials related to Hegazi-Mansour’s q-Bernoulli polynomials.
2 q-Euler numbers and polynomials
We consider the new q-extension of Euler polynomials which are generated by the generating function to be
In the special case, , are called the n th q-Euler numbers. From (2.1), we note that
By (2.1), we easily get
For example, , , , . From (1.15) and (2.1), we have
Thus, by (1.13) and (2.5), we get
Indeed, by (1.9), we get
From (2.4), we have
Thus, by (2.7) and (2.8), we get
This is equivalent to
Therefore, by (2.10), we obtain the following lemma.
Lemma 2.1 For , we have
From (2.2) we have
Thus, by (2.11), we get
Therefore, by (2.12), we obtain the following theorem.
Theorem 2.2 For , we have
For , let us assume that
Then, by (2.4), we get
From (2.14) and (2.15), we can derive the following equation:
Thus, by (2.16), we get
Therefore, by (2.14) and (2.17), we obtain the following theorem.
Theorem 2.3 For , let . Then we have
From (1.5), we note that
Let us take . Then can be represented as a linear combination of as follows:
From (1.5), we can derive the following recurrence relation for the q-Bernoulli numbers:
Thus, by (2.21), we get
For example, , , .
By (2.19), (2.20) and (2.22), we get
Therefore, by (2.23), we obtain the following theorem.
Theorem 2.4 For , we have
For , the q-Euler polynomials, , of order r are defined by the generating function to be
In the special case, , are called the n th q-Euler numbers of order r.
Then is an invertible series. From (2.24) and (2.25), we have
By (2.26), we get
Thus, by (2.26), (2.27) and (2.28), we see that
By (1.9) and (2.24), we get
Thus, we have
By (2.30), we easily get
Therefore, by (2.31) and (2.32), we obtain the following theorem.
Theorem 2.5 For , we have
Let us take . Then, by Theorem 2.3, we get
From (2.24), we have
By comparing the coefficients on the both sides of (2.35), we get
Therefore, by (2.33), (2.34) and (2.36), we obtain the following theorem.
Theorem 2.6 For , , we have
Let us assume that
By (2.29) and (2.37), we get
From (2.38), we have
Therefore by (2.37) and (2.39), we obtain the following theorem.
Theorem 2.7 For , let .
Then we have
Let us take . Then, by Theorem 2.7, we get
Therefore, by (2.40) and (2.41), we obtain the following theorem.
Theorem 2.8 For , we have
For , let us consider q-Bernoulli polynomials of order r which are defined by the generating function to be
In the special case, , are called the n th q-Bernoulli numbers of order r. By (2.42), we easily get
Let us take . Then, by Theorem 2.7, we get
Therefore, by (2.44) and (2.45), we obtain the following theorem.
Theorem 2.9 For , we have
Remark Recently, Aral, Gupta and Agarwal introduced many interesting properties and applications of q-calculus which are related to this paper (see ).
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This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MOE) (No. 2012R1A1A2003786). The authors express their sincere gratitude to the referees for their valuable suggestions and comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.
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Kim, D.S., Kim, T. Some identities of q-Euler polynomials arising from q-umbral calculus. J Inequal Appl 2014, 1 (2014). https://doi.org/10.1186/1029-242X-2014-1
- Formal Power Series
- Euler Number
- Euler Polynomial
- Vector Space Isomorphism
- Interesting Identity