Some identities of q-Euler polynomials arising from q-umbral calculus

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 [1]). In this paper, we develop the new method of q-umbral calculus due to Roman and we study 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.


Introduction
Throughout this paper we will assume q to be a fixed real number between 0 and 1. We define the q-shifted factorials by (1 − aq i ).
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 [2] defined a q-extension of Euler polynomials. In a recent paper (see [7]), B. A. Kupershmidt constructed reflection symmetries of q-Bernoulli polynomials which differ from Carlitz's q-Bernoulli numbers and polynomials. By using the method of B. A. Kupershmidt, Hegazi and Mansour also introduced new q-extension of Bernoulli numbers and polynomials (see [3,7,8]). 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-extension of Bernoulli and Euler polynomials (see [1][2][3][4][5][6][7][8][9][10]). Let C be the complex number field and let F be the set of all formal power series in variable t over C with (1.7) Let P = C[t] and let P * be the vector space of all linear functionals on P. L|p(x) denotes the action of linear functional L on the polynomial p(x), and it is well known that the vector space operations on P * are defined by where c is complex constant (see [1,5,12]).
[k]q ! t k ∈ F , we define the linear functional on P by setting f (t)|x n = a n for all n ≥ 0.
Let us assume that f L (t) = ∞ k=0 L|x n t k k! . Then by (1.9), we easily see that f L (t)|x n = L|x n . That is, f L (t) = L. Additionally, the map L −→ f L (t) is a vector space isomorphism from P * 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 f (t) of F will be thought 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 e q (yt)|x n = y n and so e q (yt)|p(x) = p(y) for p(x) ∈ P. The order o(f (t)) of the power series f (t) = 0 is the smallest integer for which a k does not vanish. If o(f (t)) = 0, then f (t) is called an invertible series. If o(f (t)) = 1, thenf (t) is called a delta series (see [1,5,11,12]).
Then we have (see [11]). (1.10) From (1.10), we have By (1.11), we get Thus from (1.12), we note that For h(t) ∈ F and p(x) ∈ P, we have and [11,12]). 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 [1]). 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 new q-extension of Euler numbers and polynomials which are derived from q-umbral calculus. Finally, we give new expicit formulas on q-Euler polynomials relate to Hegazi-Mansour's q-Bernoulli polynomials.

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, x = 0, E n,q (0) = E n,q are called the n-th q-Euler numbers. From (2.1), we note that By (2.1), we easily get , · · · . From (1.15) and (2.1), we have and 2 e q (t) + 1 x n = E n,q (x), (n ≥ 0). (2.5) Thus, by (1.13) and (2.5), we get Indeed, by (1.9), we get (2.9) This is equivalent to − 2E n−k,q = n−k−1 l=0 n − k l q E l,q , where n, k ∈ Z ≥0 with n > k.
For p(x) ∈ P n , let us assume that (2.16) Thus, by (2.16), we get where p (k) (x) = D k q p(x). Therefore, by (2.14) and (2.17), we obtain the following theorem.
From (1.5), we note that Let us take p(x) = B n,q (x) ∈ P n . Then B n,q (x) can be represented as a linear combination of {E 0,q (x), E 1,q (x), · · · , E n,q (x)} as follows: (2.20) From (1.5), we can derive the following recurrence relation for the q-Bernoulli numbers: (2.21) Thus, by (2.21), we get Therefore, by (2.23), we obtain the following theorem.
Theorem 2.4. For n ≥ 2, we have For r ∈ Z ≥0 , the q-Euler polynomials, E (r) n,q (x), of order r are defined by the generating function to be 2 e q (t) + 1 r e q (xt) = 2 e q (t) + 1 × · · · × 2 e q (t) + 1 (2.24) In the special case, n,q are called the n-th q-Euler numbers of order r.
(2.25) Then g r (t) is an invertible series. From (2.24) and (2.25), we have x n t n [n] q ! .
where p (k) (x) = D k q p(x). Let us take p(x) = E n,q (x) ∈ P n . Then, by Theorem 2.7, we get (2.42)