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

Ugur Duran, Mehmet Acikgoz


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.


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

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