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In the paper, we introduce an analogue of Haar distribution based on ( ρ , q ) -numbers, as follows: μ ρ , q ( a + p N Z p ) = ρ p N [ p N ] ρ , q ( q ρ ) a . By means of this distribution, we derive ( ρ , q ) -analogue of Volkenborn integration which is a new generalization of Kim's q -Volkenborn integration defined in [11] . From this definition, we investigate some properties of Volkenborn integration based on ( ρ , q ) -numbers. Finally, we construct ( ρ , q ) -Bernoulli numbers and polynomials derived from ( ρ , q ) -Volkenborn integral and obtain some their properties. [ABSTRACT FROM AUTHOR]

Recently, Sofonea (Gen. Math. 16:47-54, 2008) considered some relations in the context of quantum calculus associated with the q-derivative operator $D_{q}$ and divided difference. As applications of the post-quantum calculus known as the $(p,q)$ -calculus, we derive several relations involving the $(p,q)$ -derivative operator and divided differences. [ABSTRACT FROM AUTHOR]

By using the modified Milne-Thomson's polynomial given in Araci et al. (Appl Math Inf Sci 8(6):2803-2808, ), we introduce a new concept of the Apostol Hermite-Genocchi polynomials. We also perform a further investigation for aforementioned polynomial and derive some implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method. The results obtained here are an extension of Hermite-Bernoulli polynomials (Pathan and Khan in Mediterr J Math 12:679-695, ) and Hermite-Euler polynomials (Pathan and Khan in Mediterr J Math , doi:) to Apostol type Hermite-Genocchi polynomials defined in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce q -Frobenius–Euler numbers and derive some new properties. From those properties, we show that this number is a p -adic integer, and can be expressed by von Staudt–Clausen's theorem. We also get a type of Kummer congruence for this number. [ABSTRACT FROM AUTHOR]

In this paper, we present a further investigation for the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. By making use of the generating function methods and summation transform techniques, we establish some new identities involving the products of the Apostol–Bernoulli polynomials and the Apostol–Genocchi polynomials. Many of the results presented here are the corresponding generalizations of some known formulas on the classical Bernoulli polynomials and the classical Genocchi polynomials. [ABSTRACT FROM AUTHOR]