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In this paper, we derive a Mittag-Leffler function for real index and establish solutions of special type of fractional order differential equations (FDEs). The same concept is extended to discrete case by replacing polynomials into factorial polynomials and differentiation into l-difference operator. Moreover, numerical examples of our results are stated to validate our findings. The acquired results here have the ability to generate a wide range of formulas in relation to newer results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new function class called n-fractional polynomial convex function and their some algebric properties. We obtain some refinements of the right-hand side of Hermite-Hadamard inequality for the class of functions whose derivatives in absolutely value at certain powers are n-fractional polynomial convex. Also, we compare the results obtained with both Hölder, Hölder-İşcan inequalities and power-mean, improved-power-mean integral inequalities and show that the result obtained with Hölder-İşcan and improved power-mean inequalities give better approach than the others. Some applications to special means of real numbers are also given. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose a new faster iterative method, called the MNiteration process, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the MN-iteration of continuous functions on an arbitrary interval is established. We also compare the rate of convergence between the proposed iteration and some other iteration processes in the literature. Specifically, our main result shows that MN-iteration converges faster than NSP-iteration to the fixed point. We finally give numerical examples to compare the result with Mann, Ishikawa, Noor, SP and NSP iterations. Our findings improve corresponding results in the contemporary literature. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce new class of 2D-k-Laguerre-Konhauser polynomials, δL k,n(α,β) (x, y), which generalizes the 2D-Laguerre-Konhauser polynomials (see [18]). Then, we define a new family of bivariate k-Mittag-Leffler functions E k,α,β,δ(γ)(x, y) and establish the k-Riemann-Liouville double fractional integral and derivative of the functions E(γ) k,α,β,δ(x, y). Moreover, we introduce an integral operator kε(γ) α,β,δ;ω1,ω2;a+,c+ which contains the bivariate k-Mittag-Leffler functions E(γ) k,α,β,δ(x, y) in the kernel and investigate the semigroup property of this operator. Finally, the left inverse operator of the integral operator kε(γ) α,β,δ;ω1,ω2;a+,c+ is constructed. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S be a multiplicatively closed subset of R, and M be an R-module. The aim of this paper is to introduce the notion of S-copure submodules and investigate some properties of this class of modules. We say that a submodule N of M is S-copure if there exists an s ∈ S such that s(N :M I) ⊆ N + (0 :M I) for every ideal I of R. [ABSTRACT FROM AUTHOR]

In this paper, we define Gaussian Fibonacci quaternion polynomials and Gaussian Lucas quaternion polynomials. We also investigate some properties of these quaternion polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a parametric type of Cauchy polynomials and study their characteristic and combinatorial properties. In particular, we show some determinant expressions. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce and study the concepts of statistical convergence and statistical summability for martingale sequences of random variables via deferred Cesaro ` mean. We then establish an inclusion theorem concerning the relation between these two beautiful and potentially useful concepts. Also, based upon our proposed notions, we state and prove new Korovkin-type approximation theorems with algebraic test functions for a martingale sequence over a Banach space. Moreover, we demonstrate that our theorems effectively extend and improve most (if not all) of the previously existing results (in statistical and classical versions). Finally, by using the generalized Bernstein polynomials, we present an illustrative example of a martingale sequence in order to demonstrate that our established theorems are stronger than their traditional and statistical versions. [ABSTRACT FROM AUTHOR]

Consider a simple connected graph G = (V(G),E(G)), where V(G) represents the vertex set and E(G) represents the edge set respectively. A subset W of V(G) is called a resolving set for a graph G if for every two distinct vertices x, y ∈V(G), there exist some vertex w ∈W such that d(x,w) 6= d(y,w), where d(u, v) denotes the distance between vertices u and v. A resolving set of minimal cardinality is called a metric basis for G and its cardinality is called the metric dimension of G, which is denoted by β(G). A subset D of V(G) is called a doubly resolving set of G if for every two distinct vertices x, y of G, there are two vertices u, v ∈ D such that d(u, x) - d(u, y) 6= d(v, x) - d(v, y). A doubly resolving set with minimum cardinality is called minimal doubly resolving set. This minimum cardinality is denoted by ψ(G). In this paper, we determine the minimal doubly resolving sets for antiprism graphs denoted by An with n ≥ 3 and for Mobius ladders denoted by Mn, for every even positive integer n ≥ 8. It has been proved that ψ(An) = 3 for n ≥ 3 and ψ(Mn) = ( 3, if n ≡ 0 or 4 (mod 8) 4, if n ≡ 2 or 6 (mod 8) for every even positive integer n ≥ 8. [ABSTRACT FROM AUTHOR]