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Relations on the Apostol Type (p, q)-Frobenius-Euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems

Burak Kurt
Turkish Journal of Analysis and Number Theory. 2017, 5(4), 126-131. DOI: 10.12691/tjant-5-4-2
Received December 22, 2016; Revised April 19, 2017; Accepted June 14, 2017

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 (pq)-Frobenius-Euler polynomials. Also, we prove some explicit expressions.

Definition 1. Let The (pq)-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

[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
 

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Normal Style
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. http://pubs.sciepub.com/tjant/5/4/2
MLA Style
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.
APA Style
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.
Chicago Style
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