The Kronecker product is widely utilized to construct higher-dimensional spaces from lower-dimensional ones, making it an indispensable tool for efficiently analyzing multi-dimensional systems across various fields. This paper investigates the representation of analytic functions within hyper-elliptical regions through infinite series expansions involving sequences of Kronecker product bases of polynomials. Additionally, we examine the growth order and type and $T_{\rho }$-property of series composed of Kronecker product bases that represent entire functions. We also delve into the convergence properties of Kronecker product bases associated with special functions, including Bessel, Chebyshev, Bernoulli, Euler, and Gontcharoff polynomials. The obtained results extend and enhance the existing findings of such representations in hyper-spherical regions.

The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the $T_{\rho }$-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Clifford settings.

In the present paper, we establish the incomplete exponential type (IEF) of $R$-matrix functions and identify some properties of the incomplete exponential matrix functions including integral representation, some derivative formula and generating functions of the incomplete exponential of $R$-matrix functions. Finally, special cases of the presented results are pointed out.

The goal of this manuscript is to create a new faster iterative algorithm than the previous writing's sober algorithms. In the setting of Banach spaces, this algorithm is used to analyze convergence, stability, and data-dependence results. Basic numerical examples are also provided to highlight the behavior and effectiveness of our approach. Ultimately, the proposed approach is used to solve the functional Volterra-Fredholm integral problem as an application.

For a simple connected graph $G=(V,E)$, a vertex $x\in V$ distinguishes two elements (vertices or edges) $x_1\in V,y_1\in E$ if $d(x,x_1)\ne d(x,y_1).$ A subset $Q_m\subset V$ is a mixed metric generator for $G,$ if every two distinct elements (vertices or edges) of $G$ are distinguished by some vertex of $Q_m.$ The minimum cardinality of a mixed metric generator for $G$ is called the mixed metric dimension and denoted by $di{m}_m(G).$ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.

Gegenbauer polynomials hold a significant role in the constructive theory of spherical functions, while the Mittag-Leffler function is widely used in fractional calculus. In this paper, we introduce a new class of Mittag-Leffler-Gegenbauer polynomials (MLGPs) by convolutionally combining the classical Hermite polynomials with the Mittag-Leffler function of three parameters. We explore some of its aspects, such as symbolical identities, recurrence relations, differential equations, generating functions, integral representations, finite summations, and Rodrigues-type and orthogonal formulas. Additionally, we demonstrate the relevance of the MLGPs by developing and solving a fractional kinetic equation associated with the MLGPs in the kernel. Finally, employing Saigo fractional-type operators, we establish fractional integrals and derivatives formulae for our innovative MLGPs. We conclude by proposing an open question regarding the Hermite numbers and their umbral calculus for further discussion in the field of this study.

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the ${\mathbb{T}}_{\rho }$-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results.

In this study, we introduced and investigated new classes of $k$-Mittag-Leffler-Konhauser polynomials and bivariate $k$-Mittag-Leffler functions, which encompass several well-known 2D polynomials and Mittag-Leffler functions as special cases. We explored their key properties, including double $k$-Riemann-Liouville fractional calculus, double Laplace transforms, and $k$-fractional calculus operators. To further support the theoretical results, a new section presents numerical validation through simulation examples, illustrating the accuracy and practical applicability of the proposed formulas. We conclude by proposing several open questions to inspire further research and continued exploration in this area.

In this paper, we employed the $q$-Bessel Tricomi functions of zero-order to introduce bivariate extended $q$-Laguerre-based Appell polynomials. Then, the bivariate extended $q$-Laguerre-based Appell polynomials were established in the sense of quasi-monomiality. We examined some of their properties, such as $q$-multiplicative operator property, $q$-derivative operator property and two $q$-integro-differential equations. Additionally, we acquired $q$-differential equations and operational representations for the new polynomials. Moreover, we drew the zeros of the bivariate extended $q$-Laguerre-based Bernoulli and Euler polynomials, forming 2D and 3D structures, and provided a table including approximate zeros of the bivariate extended $q$-Laguerre-based Bernoulli and Euler polynomials.