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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]

Recently, the von Staudt–Clausen’s theorem for q -Euler numbers was introduced by Kim (2013) and q -Genocchi numbers were constructed by Araci et al. (2013). In this paper, we give the corresponding von Staudt–Clausen’s theorem for q -Genocchi numbers and also get the Kummer-type congruence for q -Genocchi numbers. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to deal with applications of umbral calculus on fermionic p-adic integral on . From those applications, we derive some new identities on Genocchi numbers and polynomials. Moreover, a systemic study of the class of Sheffer sequences on the generating function of Genocchi polynomials are given. [Copyright &y& Elsevier]

The significance of multi-variable special polynomials has been identified both in mathematical and applied frameworks. The article aims to focus on a new class of 3-variable Legendre-truncated-exponential-based Sheffer sequences and to investigate their properties by means of Riordan array techniques. The quasi-monomiality of these sequences is studied within the context of Riordan arrays. These sequences are expressed in determinant forms by utilizing the relation between the Sheffer sequences and Riordan arrays. Certain special Sheffer sequences are used to construct the Legendre-truncated-exponential-based Chebyshev polynomials. The same polynomials are used as base to introduce the hybrid classes involving the exponential polynomials and the Euler polynomials of higher order as illustrative examples of the aforementioned general class of polynomials. The shapes are shown and zeros are computed for these sequences by using mathematical software. The main purpose is to demonstrate the advantage of using numerical investigation and computations to discover fascinating pattern of scattering of zeros through graphical representations. [ABSTRACT FROM AUTHOR]