Abstract

In this work, we define and introduce a new kind of the Apostol type Frobenius-Euler polynomials based on the (p, q)-calculus and investigate their some properties, recurrence relationships and so on. We give some identities at this polynomial. Moreover, we get (p, q)-extension of Carlitz’s main result in [1].

1. Introduction, Definitions and Notations

Throughout this paper, we always make use of the following notation; denotes the set of natural numbers, denotes the set of nonnegative integers, denotes the set of real numbers and denotes the set of complex numbers.

The (p, q)-numbers are defined by

which is natural generalization of the q-number such that

Note that (p, q)-number is symmetric: that is

The (p, q)-derivative of a function f is defined by

The (p, q)-Gauss Binomial formula is defined by

where the notations ((p, q)-Gauss Binomial coefficients) and ((p, q)-factorial) are defined by

The (p, q)-exponential functions, and are defined by

and

From this form, we easily see that

In this work, we introduce Apostol type (p, q)-Frobenius-Euler polynomials. We give some new identities for the Apostol type (p, q)-Frobenius-Euler polynomials. Also, we prove some explicit expressions.

Definition 1. Let The (p, q)-Bernoulli numbers and polynomials are defined by means of the generating functions in ⁶:

Definition 2. Let The (p, q)-Euler numbers and polynomials are defined by means of the generating functions in ⁶:

Definition 3. Let The (p, q)-Bernoulli numbers and polynomials in x, y of order are defined by means of the generating functions in ⁶:

(1.1)

(1.2)

Definition 4. Let The (p, q)-Euler numbers and polynomials in x, y of order are defined by means of the generating functions in ⁶:

(1.3)

(1.4)

Classical Frobenius-Euler polynomials of order is defined by the following relation ^{1, 7, 10, 11}.

(1.5)

where u algebraic number.

Similarly Frobenius-Euler polynomials of order is defined by the following relation ( ¹⁷)

(1.6)

Definition 5. The Apostol type q-Frobenius-Euler polynomials of order in x, y and Apostol type q-Frobenius-Euler number of order in ⁹ respectively

Definition 6. Let and . We define the Apostol type (p, q)-Bernoulli polynomials of order in x, y and the Apostol type (p, q)-Bernoulli numbers of order in x, y respectively

Definition 7. Let and We define the Apostol type (p, q)-Euler polynomials of order in x, y and the Apostol type (p, q)-Euler numbers of order in x, y respectively

Definition 8. We de.ne Apostol type (p, q)-Frobenius-Euler polynomials of order in x, y and Apostol type (p, q)-Frobenius-Euler numbers of order respectively

(1.7)

(1.8)

Letting in (1.7), we have

⁹.

Putting and in (1.7), we have

where is -Euler polynomials of order .

Using in last equation, we have

where is -Euler polynomials of order

Letting in last equation, we have

where is Hermite based Euler polynomials of order

2. Some Basic Properties for the Apostol Type q-Frobenius-Euler Polynomials

Proposition 1. Apostol type Frobenius-Euler polynomials satisfy the following relations

(2.1)

(2.2)

(2.3)

Theorem 1. For and the following relationships hold true:

(2.4)

Proof. Using Definition

Comparing the coefficients of we have (2.4). Similarly the other equation is been calculation.

Theorem 2. There is the following relation for the generalized Apostol type q-Frobenius-Euler polynomials

(2.5)

Proof. By using the identity

Comparing the coefficient of , we prove (2.5).

Remark 1. For Substituting in (2.5). We have Carlitz result ( ¹, equation 2.19).

Theorem 3. There is the following relation for the generalized Apostol type (p, q)-Frobenius-Euler polynomial

(2.6)

Proof. By using the identity

We write as

Comparing the coefficients of we have

3. Explicit Relation for the Apostol Type (p, q)-Frobenius-Euler Polynomials

Theorem 4. There is the following relation for the Apostol type (p, q)-Frobenius-Euler poly-nomials

(3.1)

Proof. Since (1.7);

Comparing the coefficients of , we have (3.1).

Theorem 5. There is the following relation between Apostol type (p, q)-Frobenius-Euler polynomials and the generalized Apostol (p, q)-Bernoulli polynomials

(3.2)

Proof.

Comparing the coefficients of , we have (3.2).

Corollary 1. There is the following relation between Apostol type (p, q)-Frobenius-Euler polynomials and the generalized Apostol (p, q)-Euler polynomials

Acknowledgements

The present investigation was supported, by the Scientific Research Project Administration of Akdeniz University.

References