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This Editorial introduces the Symmetry Special Issue “Theory and Applications of Special Functions II” and summarizes the nine contributions collected therein. The papers span the analytic continuation of multivariate hypergeometric functions; stability theory for differential equations via integral transforms; numerical schemes for multi-space fractional partial differential equations based on nonstandard finite differences and orthogonal polynomials; applications of the Lambert W function to viscoelastic creep modeling; algebraic constructions of new Hermite-type polynomial families via the monomiality principle; higher-level generalizations of poly-Cauchy numbers; Bell-polynomial expansions for Laplace transforms of higher-order nested functions; and two complementary studies on the physical implementation and algebraic description of Gaussian quantum states. Beyond the contributions of the Special Issue, we highlight methodological connections—continued fractions and complex analysis, transform techniques, special polynomials, and combinatorial sequences—and emphasize the unifying role of symmetry across mathematical structures and applications.

We review dielectric resonator antenna (DRA) designs. This review examines recent advancements across several categories, specifically focusing on their applicability in array configurations for millimeter-wave (mmW) bands, particularly in the context of 5G and beyond 5G applications. Notably, the off-chip DRA designs, including in-substrate and compact DRAs, have gained prominence in recent years. This surge in popularity can be attributed to the rapid development of cost-effective multilayer laminate manufacturing techniques, such as printed circuit boards (PCBs) and low-temperature co-fired ceramic (LTCC). Furthermore, there is a growing demand for DRAs with beam-steering, dual-band functions, and on-chip alignment availability, as they offer versatile alternatives to traditional lossy printed antennas. DRAs exhibit distinct advantages of lower conductive losses and greater flexibility in shapes and materials. We discuss and compare the performances of different DRA designs, considering their material usage, manufacturing feasibility, overall performance, and applications. By exploring the pros and cons of these diverse DRA designs, this review provides valuable insights for researchers in the field.

The fractional exponential function is considered. General expansions in fractional powers are used to solve fractional population dynamics problems. Laguerretype exponentials are also considered, and an application to Laguerre-type fractional logistic equation is shown.

This paper explores the eigenfunctions of specific Laguerre-type parametric operators to develop multi-parametric models, which are associated with a class of the generalized Mittag-Leffler type functions, for dynamical systems and population dynamics. By leveraging these multi-parametric approaches, we introduce new concepts in number theory, specifically those involving multi-parametric Bernoulli and Euler numbers, along with other related polynomials. Several numerical examples, which are generated by using the computer algebra program Mathematica© (Version 14.3), demonstrate the effectiveness of the models that we have presented and analyzed in this paper.

Electronically controlled antenna arrays, such as reconfigurable and phased antenna arrays, are essential elements of high-frequency 5G communication hardware. These antenna arrays are aimed at delivering specified communication scenarios and channel characteristics in the mm-wave parts of the 5G spectrum. At the same time, several challenges are associated with the development of such antenna structures, and these challenges mainly originate from their intended mass production, contemporary manufacturing technologies, integration with active RF chains, compact size, dense circuitry, and limitations in postmanufacturing tuning. Consequently, 5G antenna array designers are presented with contradictory design requirements and constraints. Furthermore, these designers need to handle large numbers of designable parameters of the antenna array models, which can be computationally expensive, especially for repetitive and adaptive simulations that are required in design optimization and tuning. Antenna array synthesis, namely, the process of finding positions, orientation, and excitation of the array radiators, is a challenging yet crucial part of antenna array development. This process ensures that the performance requirements of the antenna array are met. Therefore, there is a need for reliable yet fast automated computer-aided design (CAD) and synthesis tools that can support the development of 5G antenna array solutions, from the initial prototyping stage to the final manufacturing tolerance analysis. This paper presents an overview of recent advances in antenna array synthesis from the viewpoint of their applicability to the design of electronically reconfigurable and phased antenna arrays for wireless communications and remote sensing.

A novel linear active rampart array that is able to steer in the azimuth plane through frequency sweeping and in the elevation plane through phase shifting has been developed. The presented system uses eight active elements and two parasitic ones to ensure consistent radiation patterns. It achieves an impedance bandwidth of 15% centered around 8.625 GHz and has a field of view of 140° in both scanning planes. The ultrawide scan range through frequency sweeping is achieved by feeding each antenna element from two different ports. By feeding the rampart antenna from one port, the radiation pattern can be steered in the subspace from −70° to 0°, and when feeding from the other port, the radiation pattern can be steered in the subspace from 0° to +70°. An improved mathematical model was used to predict the influence of the design parameters of a rampart antenna on its radiation properties. The results obtained by the mathematical model were compared with full‐wave solutions, and a software tool was crafted with the aim to streamline the design process of rampart antennas.

We extended the classical Bernoulli and Euler numbers and polynomials to introduce the Laguerre-type Bernoulli and Euler numbers and related fractional polynomials. The case of fractional Bernoulli and Euler polynomials and numbers has already been considered in a previous paper of which this article is a further generalization. Furthermore, we exploited the Laguerre-type fractional exponentials to define a generalized form of the classical Laplace transform. We show some examples of these generalized mathematical entities, which were derived using the computer algebra system Mathematica© (latest v. 14.0).

Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order.

Fractional derivative operators are finding applications in a wide variety of fields with their ability to better model certain phenomena exhibiting spatial and temporal nonlocality. One area in which these operators are applicable is in the field of electromagnetism, thereby modelling transient wave propagation in complex media. To apply fractional derivative operators to electromagnetic problems, the operator must adhere to certain principles, like the trigonometric functions invariance property. The Grünwald–Letnikov and Marchaud fractional derivative operators comply with these principles and therefore could be applied. The fractional derivative arises when modelling frequency-dispersive dielectric media. The time-domain convolution integral in the relation between the electric displacement and the polarisation density, containing an empirical extension of the Debye model, is approximated directly. A common approach is to recursively update the convolution integral by approximating the time series by a truncated sum of decaying exponentials, with the coefficients found through means of optimisation or fitting. The finite-difference time-domain schemes using this approach have shown to be more computationally efficient compared to other approaches using auxiliary differential equation methods.

Bell’s polynomials are used in many different fields of mathematics, ranging from number theory to operator theory. This paper shows a relevant application in probability theory aimed at computing the moments of generalized Gaussian distributions. To this end, a table containing the first values of the complete Bell’s polynomials is provided. Furthermore, a dedicated code for approximating the moments of the general distributions in terms of complete Bell’s polynomials is detailed. Several test cases concerning different nested functions are discussed.