Apostol Type \((p, q)\)-Bernoulli, \((p, q)\)-Euler and \((p, q)\)-Genocchi Polynomials and Numbers

Authors

  • Ugur Duran Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep
  • Mehmet Acikgoz Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep

DOI:

https://doi.org/10.26713/cma.v8i1.512

Keywords:

\((p, q)\)-calculus, Apostol Bernoulli polynomials, Apostol Euler polynomials, Apostol Genocchi polynomials, Generating function, Cauchy product

Abstract

The main subject of this work is to introduce and investigate a new generalizations of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials under the theory of post quantum calculus, denoted by \((p, q)\)-calculus. We call them Apostol type \((p, q)\)-Bernoulli polynomials of order \(\alpha\), Apostol type \((p, q)\)-Euler polynomials of order \(\alpha\) and the Apostol type \((p, q)\)-Genocchi polynomials of order \(\alpha\). We derive some of their properties involving addition theorems, difference equations, derivative properties, recurrence relationships, and so on. Also, \((p, q)\)-analogues of some familiar formulae belonging to usual Apostol-Bernoulli, Euler and Genocchi polynomials are shown. Furthermore, \((p, q)\)-generalizations of Cheon's main result [G.S. Cheon, Appl. Math. Lett. 16 (2003) 365–368] and the formula of Srivastava and Pintér [H.M. Srivastava, A. Pintér, Appl. Math. Lett. 17 (2004), 375–380] are investigated.

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References

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Published

30-06-2017
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How to Cite

Duran, U., & Acikgoz, M. (2017). Apostol Type \((p, q)\)-Bernoulli, \((p, q)\)-Euler and \((p, q)\)-Genocchi Polynomials and Numbers. Communications in Mathematics and Applications, 8(1), 7–30. https://doi.org/10.26713/cma.v8i1.512

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Research Article