By using the modified Milne-Thomson’s polynomial given in Araci et al. (Appl Math Inf Sci 8(6):2803–2808, 2014), 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, 2015a) and Hermite–Euler polynomials (Pathan and Khan in Mediterr J Math 2015b, doi: 10.1007/s00009-015-0551-1 ) to Apostol type Hermite–Genocchi polynomials defined in this paper.

The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a trans-Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. We provide numerous physical applications of the derived inequalities. Finally, we prove that under a certain condition the base manifold is isometric to a sphere with a constant sectional curvature.

In this article, we generalize fractional operators (differential and integral) in the unit disk. These operators are generalized the Srivastava-Owa operators. Geometric properties are studied and the advantages of these operators are discussed. As an application, we impose a method, involving a memory formalism of the Beer-Lambert equation based on a new generalized fractional differential operator. We give solutions in terms of the multi-index Mittag-Leffler function. In addition, we sanctify the out come from a stochastic standpoint. We utilize the generalized Wright function to obtain the analytic formula of solutions.

In this paper, we discuss open (closed) sets, rough interiors, rough closures, continuous mappings, open (closed) mappings and homeomorphism mappings of covering based rough topology of Akduman, Özcelik and Özel. We also construct the nano topology on a given covering based rough set.

In this paper, we introduce a rough quotient. Also, we present conditions ensuring that G/H are partitions of G. The rough projection map is also presented. We discuss first, second and third rough isomorphism theorems and other related results. At the end, an orbit and a stabilizer in topological rough groups are considered.

This paper examines the topological features of the compact biwarped product submanifolds of a space form with vanishing constant sectional curvature. More precisely, we show that stable integral p−current does not exist in a compact oriented biwarped product submanifold in an Euclidean space that meets some geometric conditions based on Laplacian of warping functions, slant functions. Simultaneously, it is shown that their homology group are zero under these geometric conditions. Additionally, some special cases are also described.