On Generalized Some (p, q)-Special Polynomials

Corresponding Author: Serkan Araci Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey Email: mtsrkn@hotmail.com Abstract: In this study, we introduce a new class of generalized (p, q)Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials and investigate their some properties. We derive (p, q)-generalizations of some familiar formulae belonging to classical Bernoulli, Euler and Genocchi polynomials. We also obtain a (p, q)-extension of the Srivastava-Pintér addition theorem.


Introduction
Throughout of the paper, N denotes the set of the natural numbers, 0 N denotes the set of nonnegative integers, R denotes the set of real numbers and C denotes the set of complex numbers.
In the next section, we introduce a new class of generalized (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials of order α. Then we investigate addition theorems, difference equations, derivative properties, integral representations, recurrence relationships for aforementioned polynomials. Also, we derive (p, q)-analogues of some known formulae belonging to usual Bernoulli, Euler and Genocchi polynomials. Further-more, we discover (p, q)analogue of the main results given earlier by Srivastava and Pintér (2004).
Generalized (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi Polynomials We consider a new approach to higher order Bernoulli, Euler and Genocchi polynomials and numbers in the light of (p, q)-calculus. We firstly state the following Definition 1.

Definition 1
For p, q, α ∈ C with 0<|q| <|p|<1 and m ∈ N , the generalized (p, q)-Bernoulli polynomials [ Upon setting x = 0 and y = 0 in Definition 1, we then which are called, respectively, n-th generalized (p, q)-Bernoulli number of order α, n-th generalized (p, q)-Euler number of order α and n-th generalized (p, q)-Genocchi number of order α. In the case α = 1 in Definition 1, we have:

Remark 1
The order α of the Apostol type (p, q)-polynomials in Definition 1 (and also in all analogous situations occuring elsewhere in this paper) is tacitly assumed to be a nonnegative integer except possibly in those cases in which the right-hand side of the generating functions (2.1), (2.2) and (2.3) turns out to be a power series in z. Only in these latter cases, we can safely assume that α ∈ C .

Remark 2
Putting m = 1 in Definition 1 reduces to (p, q)analogue of Bernoulli ( ) n α B (x, y: p, q), Euler ( ) n α E (x, y: p, q) and Genocchi polynomials ( ) n α G (x, y: p, q) defined in (Duran et al., 2016b), as follows: We now give some special cases of Definition 1 as Corollary 3 and Corollary 4 as follows.

Remark 3
If we take p = 1 in Definition 1, we then get (Mahmudov and Keleshteri, 2014;: called n-th generalized q-Bernoulli polynomial of order α, n-th generalized q-Euler polynomial of order α and n-th generalized q-Genocchi polynomial of order α, respectively.

Remark 4
Taking q→1, p = α = m = 1 and y = 0 in 1 gives: which are known as classical Bernoulli polynomials, classical Euler polynomials and Genocchi polynomials, respectively, (Kurt, 2013;Mahmudov and Keleshteri, 2014;Ozden, 2010;Tremblay et al., 2011). We now discuss some properties and behaviours of the aforementioned polynomials in Definition 1. We first provide the following basic properties as Proposition 1 without proving, because they can be obtained by using Definition 1 and Cauchy product.

Proposition 1
The following relations hold true: n k n k m n k n k k p q n n k n k m k k p q n x y p q q x p q y k n p y p q x k n x y p q q x p q y k A special case of Proposition 1 is given by Corollary 1.

Proposition 4
We have: Since (Sadjang, 2013):  The other integral representations can be proved by utilizing similar proof technique used above.
We give the following recurrence relationships.

Theorem 1
We have:

Proof
By inspiring the proof technique in (Mahmudov and Keleshteri, 2013) and utilizing the following relation: Checking against the coefficients of z n , then we have the Equation 2.10. The others in this theorem can be similarly proved.
By the Equation 2.5 and 2.10, Equation 2.7 and 2.11, Equation 2.9 and 2.12, we acquire the following formulas.
Note that: n m n m m n k p q k p q n p y p q k n m y n m n q n y p q k G G From Theorem 1, we get the following Corollary 2.

Corollary 2
The following equalities:  n l x y p q y p q x p q p q l k n l y p q x p q p q l k n l x y p q y p q x p q p q l k n l y p q x p q p q l k n l x y p q y p q x p q p q l k n l y p q x p q p q l k

Proof
From Definition 1 and using Cauchy product, we get: Checking against the coefficients of z n /[n] p,q !, then we have desired result in the first equation. The others in this theorem can be proved in a like manner.
We give some relations between the new and old (p, q)-polynomials as follows.

Theorem 3
For 0 n ∈ N and x,y, α ∈ C , the following relations holds true:  x p q l p x p q s (2.14) where B n (x, y: p, q), E n (x, y: p, q) and G n (x, y: p, q) are defined in (Duran et al., 2016b), called (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials, respectively.

Proof
Indeed: Comparing the coefficients of z n /[n] p,q !, we get desired result for (2.14). The others in this theorem can be proved in a like manner. Here

Theorem 4
For 0 n ∈ N , m ∈ N and x, y, α ∈ C , the following correlations hold true:    x p q l p x p q n x y p q ly p q l u u x p q l p x p q s  Here we give the following theorem.

Theorem 5
We have: The proof of this theorem can be easily completed by using the same proof method in the proof of Theorem 4. So we omit them.
From Corollary 1 and Theorem 5, we have the following Corollary 3.

Corollary 3
We have:

Theorem 6
For 0 n ∈ N and α ∈ C , the following relationships are valid:

Conclusion
We have introduced new generalizations of Bernoulli polynomials, Euler polynomials and Genocchi polynomials which are called generalized (p, q)-Bernoulli polynomials, (p, q)-Euler polynomials and (p, q)-Genocchi polynomials of order α. We have examined their several properties and relationships including additions theorems, difference equations, differential relations, recurrence relationships and so on. Also, we have given the (p, q)-extension of the formula of Srivastava and Pintér (2004). The results derived in this paper reduce to known properties of generalized q-polynomials of order α when p = 1, (Mahmudov and Keleshteri, 2014;. Also, in the case q → p = 1, our results in this paper reduce to ordinary results for the generalized Bernoulli, Euler and Genocchi polynomials of order α.