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

Burak Kurt^1,

¹Department of Mathematics, Faculty of Educations, University of Akdeniz

Cite this paper:
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. 2017; 5(4):126-131. doi: 10.12691/tjant-5-4-2

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].

Keywords:
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.

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

References:

[1]	Carlitz, L., Eulerian numbers and polynomials, Math. Mag., 32(1959), 247-260.
[2]	Carlitz, L., q-Bernoulli numbers and polynomials, Duke Math. J., 15(1948), 987-1050.
[3]	Carlitz, L., q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., 76(1954), 332-350.
[4]	Cenkci, M., Can, M. and Kurt, V., q-extensions of Genocchi numbers, J. Korean Math. Soc., 43(2006), 183-198.
[5]	Cheon, G. S., A note on the Bernoulli and Euler polynomials, Appl. Math. Letter, 16(2003), 365-368.
[6]	Duran, U., Acikgoz, M. and Araci, S., On (p, q)-Bernoulli, (p, q)-Euler and (p, q)-Genocchi polynomials, 2016, submitted.
[7]	Kim, T., Identities involving Frobenius-Euler polynomials arising from non-linear differential equation, J. Number Theory, 132(2012), 2854-2865.
[8]	Kim, T., Some formulae for the q-Bernoulli and Euler polynomials of higher order, J. Math. Analy. Appl., 273(2002), 236-242.
[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.
[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.
[11]	Kurt, B. and Simsek Y., On the generalized Apostol type Frobenius-Euler polynomials, Adv. in Diff. Equ.
[12]	Kac, V. and Cheung, P., Quantum Calculus, Springer (2002).
[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.
[14]	Luo, Q.-M., Some results for the q-Bernoulli and q-Euler polynomials, J. Math. Anal. Appl., 363 (2010), 7-18.
[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.
[16]	Mahmudov, N. I., On a class of q-Bernoulli and q-Euler polynomials, Adv. in Diff. Equa., 2013.
[17]	Simsek, Y., Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials, Axioms, 1(2012), 395-403.
[18]	Simsek, Y., Generating functions for generalized Stirling type numbers Array type polynomials, Eulerian type polynomials and their applied, Arxiv: 1111.3848v1.2011.
[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.
[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).
[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.
[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.
[23]	Srivastava, H. M. and Choi, J., Series associated with the zeta and related functions, Kluwer Academic Publish, London (2011).
[24]	Srivastava, H. M. and Pintér, A., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Letter, 17(2004), 375-380.
[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.