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Research Article

Open Access Peer-reviewed

Burak Kurt^{ }

Received December 22, 2016; Revised April 19, 2017; Accepted June 14, 2017

Generating function Frobenius-Euler polynomials and numbers (*p** **q*)-calculus (*p** **q*)-Frobenius-Euler polynomials Apostol-Bernoulli number and polynomials generalized q-Bernoulli polynomials generalized q-Euler polynomials.

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].

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 de**fi**ned by means of the generating functions in *^{ 6}:

**Definition 2.** *Let** ** ** **The (p**,** **q)-Euler numbers** ** **and polynomials** ** **are de**fi**ned by means of the generating functions in* ^{ 6}:

**Definition 3.** *Let** ** ** **The (p, q)-Bernoulli numbers** ** **and polynomials** ** **in x, y of order** ** **are de**fi**ned by means of the generating functions in* ^{ 6}:

(1.1) |

(1.2) |

**De****fi****nition 4.*** Let** ** ** **The **(p, q)**-Euler numbers** ** **and polynomials** ** **in x, y of order** ** **are de**fi**ned 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

**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

**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*

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

[1] | Carlitz, L., Eulerian numbers and polynomials, Math. Mag., 32(1959), 247-260. | ||

In article | View Article | ||

[2] | Carlitz, L., q-Bernoulli numbers and polynomials, Duke Math. J., 15(1948), 987-1050. | ||

In article | View Article | ||

[3] | Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76(1954), 332-350. | ||

In article | |||

[4] | Cenkci, M., Can, M. and Kurt, V., q-extensions of Genocchi numbers, J. Korean Math. Soc., 43(2006), 183-198. | ||

In article | View Article | ||

[5] | Cheon, G. S., A note on the Bernoulli and Euler polynomials, Appl. Math. Letter, 16(2003), 365-368. | ||

In article | View Article | ||

[6] | Duran, U., Acikgoz, M. and Araci, S., On (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials, 2016, submitted. | ||

In article | View Article | ||

[7] | Kim, T., Identities involving Frobenius-Euler polynomials arising from non-linear differential equation, J. Number Theory, 132(2012), 2854-2865. | ||

In article | View Article | ||

[8] | Kim, T., Some formulae for the q-Bernoulli and Euler polynomials of higher order, J. Math. Analy. Appl., 273(2002), 236-242. | ||

In article | View Article | ||

[9] | Kurt, B., A Note on the Apostol type q-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pinter Addition Theorems, Filomat, 2016, 30(1), 65-72. | ||

In article | View Article | ||

[10] | Kurt, B. and Simsek Y., Frobenius-Euler type polynomials related to Hermite-Bernoulli polynomials, Numerical Analysis and Appl. Math. ICNAAM 2011 Conf. Proc., 1389(2011), 385-388. | ||

In article | View Article | ||

[11] | Kurt, B. and Simsek Y., On the generalized Apostol type Frobenius-Euler polynomials, Adv. in Diff. Equ. | ||

In article | View Article | ||

[12] | Kac, V. and Cheung, P., Quantum Calculus, Springer (2002). | ||

In article | View Article | ||

[13] | Luo, Q.-M. and Srivastava, H. M., Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comp. Math. App., 51(2006), 631-642. | ||

In article | View Article | ||

[14] | Luo, Q.-M., Some results for the q-Bernoulli and q-Euler polynomials, J. Math. Anal. Appl., 363 (2010), 7-18. | ||

In article | View Article | ||

[15] | Mahmudov, N. I., q-analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems, Discrete Dynamics in Nature and Soc., Article number 169348, 2012. | ||

In article | View Article | ||

[16] | Mahmudov, N. I., On a class of q-Bernoulli and q-Euler polynomials, Adv. in Diff. Equa., 2013. | ||

In article | View Article | ||

[17] | Simsek, Y., Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms, 1(2012), 395-403. | ||

In article | View Article | ||

[18] | Simsek, Y., Generating functions for generalized Stirling type numbers Array type polynomials, Eulerian type polynomials and their applied, Arxiv: 1111.3848v1.2011. | ||

In article | View Article | ||

[19] | Srivastava, H. M., Some generalization and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Mah. Inform. Sci., 5(2011), 390-444. | ||

In article | View Article | ||

[20] | Srivastava, H. M., Kurt, B. and Simsek, Y., Some families of Genocchi type polynomials and their interpolation function, Integral Trans. and Special func., 23(2012). | ||

In article | View Article | ||

[21] | Srivastava, H. M., Garg, M. and Choudhary, S., A new genralization of the Bernoulli and related polynomials, Russian J. Math. Phys., 17(2010), 251-261. | ||

In article | View Article | ||

[22] | Srivastava H. M., Garg M. and Choudhary S., Some new families of the generalized Euler and Genocchi polynomials, Taiwanese J. Math., 15(2011), 283-305. | ||

In article | View Article | ||

[23] | Srivastava, H. M. and Choi, J., Series associated with the zeta and related functions, Kluwer Academic Publish, London (2011). | ||

In article | View Article | ||

[24] | Srivastava, H. M. and Pintér, A., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Letter, 17(2004), 375-380. | ||

In article | View Article | ||

[25] | Trembley, R., Gaboury, S. and Fugére, B. J., A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pintér addition theorems, Appl. Math. Letter, 24(2011), 1888-1893. | ||

In article | View Article | ||

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit https://creativecommons.org/licenses/by/4.0/

Burak Kurt. Relations on the Apostol Type (*p,** **q*)-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems. *Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 4, 2017, pp 126-131. https://pubs.sciepub.com/tjant/5/4/2

Kurt, Burak. "Relations on the Apostol Type (*p,** **q*)-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems." *Turkish Journal of Analysis and Number Theory* 5.4 (2017): 126-131.

Kurt, B. (2017). Relations on the Apostol Type (*p,** **q*)-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems. *Turkish Journal of Analysis and Number Theory*, *5*(4), 126-131.

Kurt, Burak. "Relations on the Apostol Type (*p,** **q*)-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems." *Turkish Journal of Analysis and Number Theory* 5, no. 4 (2017): 126-131.

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[1] | Carlitz, L., Eulerian numbers and polynomials, Math. Mag., 32(1959), 247-260. | ||

In article | View Article | ||

[2] | Carlitz, L., q-Bernoulli numbers and polynomials, Duke Math. J., 15(1948), 987-1050. | ||

In article | View Article | ||

[3] | Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76(1954), 332-350. | ||

In article | |||

[4] | Cenkci, M., Can, M. and Kurt, V., q-extensions of Genocchi numbers, J. Korean Math. Soc., 43(2006), 183-198. | ||

In article | View Article | ||

[5] | Cheon, G. S., A note on the Bernoulli and Euler polynomials, Appl. Math. Letter, 16(2003), 365-368. | ||

In article | View Article | ||

[6] | Duran, U., Acikgoz, M. and Araci, S., On (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials, 2016, submitted. | ||

In article | View Article | ||

[7] | Kim, T., Identities involving Frobenius-Euler polynomials arising from non-linear differential equation, J. Number Theory, 132(2012), 2854-2865. | ||

In article | View Article | ||

[8] | Kim, T., Some formulae for the q-Bernoulli and Euler polynomials of higher order, J. Math. Analy. Appl., 273(2002), 236-242. | ||

In article | View Article | ||

[9] | Kurt, B., A Note on the Apostol type q-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pinter Addition Theorems, Filomat, 2016, 30(1), 65-72. | ||

In article | View Article | ||

[10] | Kurt, B. and Simsek Y., Frobenius-Euler type polynomials related to Hermite-Bernoulli polynomials, Numerical Analysis and Appl. Math. ICNAAM 2011 Conf. Proc., 1389(2011), 385-388. | ||

In article | View Article | ||

[11] | Kurt, B. and Simsek Y., On the generalized Apostol type Frobenius-Euler polynomials, Adv. in Diff. Equ. | ||

In article | View Article | ||

[12] | Kac, V. and Cheung, P., Quantum Calculus, Springer (2002). | ||

In article | View Article | ||

[13] | Luo, Q.-M. and Srivastava, H. M., Some relationships between the Apostol-Bernoulli and Apostol-Euler polynomials, Comp. Math. App., 51(2006), 631-642. | ||

In article | View Article | ||

[14] | Luo, Q.-M., Some results for the q-Bernoulli and q-Euler polynomials, J. Math. Anal. Appl., 363 (2010), 7-18. | ||

In article | View Article | ||

[15] | Mahmudov, N. I., q-analogues of the Bernoulli and Genocchi polynomials and the Srivastava-Pintér addition theorems, Discrete Dynamics in Nature and Soc., Article number 169348, 2012. | ||

In article | View Article | ||

[16] | Mahmudov, N. I., On a class of q-Bernoulli and q-Euler polynomials, Adv. in Diff. Equa., 2013. | ||

In article | View Article | ||

[17] | Simsek, Y., Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms, 1(2012), 395-403. | ||

In article | View Article | ||

[18] | Simsek, Y., Generating functions for generalized Stirling type numbers Array type polynomials, Eulerian type polynomials and their applied, Arxiv: 1111.3848v1.2011. | ||

In article | View Article | ||

[19] | Srivastava, H. M., Some generalization and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Mah. Inform. Sci., 5(2011), 390-444. | ||

In article | View Article | ||

[20] | Srivastava, H. M., Kurt, B. and Simsek, Y., Some families of Genocchi type polynomials and their interpolation function, Integral Trans. and Special func., 23(2012). | ||

In article | View Article | ||

[21] | Srivastava, H. M., Garg, M. and Choudhary, S., A new genralization of the Bernoulli and related polynomials, Russian J. Math. Phys., 17(2010), 251-261. | ||

In article | View Article | ||

[22] | Srivastava H. M., Garg M. and Choudhary S., Some new families of the generalized Euler and Genocchi polynomials, Taiwanese J. Math., 15(2011), 283-305. | ||

In article | View Article | ||

[23] | Srivastava, H. M. and Choi, J., Series associated with the zeta and related functions, Kluwer Academic Publish, London (2011). | ||

In article | View Article | ||

[24] | Srivastava, H. M. and Pintér, A., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Letter, 17(2004), 375-380. | ||

In article | View Article | ||

[25] | Trembley, R., Gaboury, S. and Fugére, B. J., A new class of generalized Apostol-Bernoulli polynomials and some analogues of the Srivastava-Pintér addition theorems, Appl. Math. Letter, 24(2011), 1888-1893. | ||

In article | View Article | ||