Special Issue Editorial: Theory and Applications of Special Functions II

Abstract

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.

1. Introduction

Special functions play a pivotal role in advanced problems of mathematical physics and engineering. They provide exact or approximate analytical solutions to complex equations, offering deep insights into underlying phenomena. Many classical special functions (e.g., hypergeometric functions, orthogonal polynomials, and Bessel functions) arise naturally in problems with specific symmetry properties, and their modern generalizations continue to find new applications. The continued vitality of this field is evident despite early predictions that computational methods might render special functions obsolete. In reality, numerical solutions often lack the analytical clarity that special function expressions afford, and the study of special functions has experienced a strong resurgence. Active research groups worldwide contribute to this area, and numerous conferences and journal issues (including a preceding installment of this Special Issue) have been devoted to recent advances in special functions and their applications. This second installment of the Special Issue, Theory and Applications of Special Functions II, presents nine original papers that illustrate the breadth of current research, spanning theoretical developments, computational techniques, and novel applications in physics and engineering.

2. Contributions of the Special Issue

The contributions in this Special Issue cover a wide spectrum of topics. On the theoretical side, one study extends the classical theory of hypergeometric functions: Dmytryshyn [1] establishes new symmetric domains for the analytic continuation of Appell’s bivariate hypergeometric function $F_2$ by employing branched continued fractions, thus broadening the function’s applicability in complex domains. Another paper focuses on a different special function: Mainardi et al. [2] explore the properties of the multivalued Lambert W function and demonstrate its application in modeling linear viscoelastic phenomena. By leveraging the Bernstein and Stieltjes positivity properties of the Lambert W function, as well as its conjugate symmetry along the negative real axis, they derive spectral and relaxation functions that describe a creep compliance model in rheology [2]. These two works exemplify how extending the analytical framework of special functions can lead to a deeper understanding of physical processes.

Several contributions address challenging problems in differential equations using special function techniques. Saad and Srivastava [3] develop a hybrid numerical scheme for a fractional-order Korteweg–de Vries system arising in multi-space physics. Their approach combines a non-standard finite difference method with a spectral collocation technique based on shifted Vieta–Lucas orthogonal polynomials, enabling accurate solutions for the coupled fractional differential equations [3]. Pinelas et al. [4] investigate the Ulam–Hyers stability of linear differential equations via an integral transform method. They introduce a general integral transform to establish criteria for two types of stability (classical Ulam–Hyers and a generalized Mittag-Leffler stability) in differential systems. As an application, they analyze the stability of a differential model in an electrical circuit and provide illustrative examples and graphical results confirming the effectiveness of the method [4]. Together, these studies [3,4] underscore the effectiveness of special-function-based techniques and transforms in solving and analyzing complex differential equations, including those of fractional order.

A significant portion of this Special Issue is devoted to advances in special polynomials and combinatorial number sequences. Dattoli and Licciardi [5] introduce an extended family of Hermite polynomials derived through the monomiality principle, an operational method that treats generalized derivative and multiplication operators abstractly. Using this framework, they unify various families of special polynomials as monomials and derive a new two-variable generalization of Hermite polynomials, along with the corresponding differential equations and orthogonality properties [5]. In a related vein, Caratelli and Ricci [6] apply Bell’s polynomials (a well-known family of combinatorial polynomials) to the analysis of nested functional compositions. They present an analytical technique to approximate the Laplace transforms of higher-order nested functions by exploiting extended Bell’s polynomials, thereby providing a systematic way to handle iterated integrals and derivatives of composite functions [6]. Komatsu and Sirvent [7] focus on special number sequences by defining and exploring poly-Cauchy numbers of higher order. Generalizing the known poly-Cauchy numbers (level 2) to an arbitrary level s, they derive explicit formulas, recurrence relations, and generating functions for these numbers. Moreover, they reveal connections of higher-level Cauchy numbers to iterated integrals and even provide determinant representations, enriching the combinatorial theory of special numbers [7]. These three papers [5,6,7] highlight the ongoing development of operational techniques and symbolic methods in the realm of special polynomials and numbers, which have implications for both pure and applied mathematics.

Finally, two contributions in this Special Issue delve into the quantum domain, examining Gaussian states from a symmetry and implementation perspective. Cariolaro and Corvaja [8] address the problem of constructing two-mode Gaussian quantum states whose covariance matrix is in the so-called standard form, a symmetric form characterized by only four independent parameters (symplectic invariants). They propose an explicit optical implementation using basic components, a sequence of beam splitters and single-mode squeezers, to realize any two-mode Gaussian state with a given standard-form covariance matrix. This construction provides clear physical interpretations for each parameter in the covariance matrix and demonstrates a non-redundant way to generate Gaussian states [8]. In a companion study, Cariolaro et al. [9] further generalize the discussion to multimode Gaussian unitaries and states. They develop an algebraic formalism to evaluate the resulting covariance matrix when implementing Gaussian transformations with primitive optical components, thereby offering a straightforward computational toolkit (free of radical expressions) for analyzing Gaussian-state operations [9]. The insights from these two works [8,9] are valuable for quantum information science, where symmetric covariance structures underlie the efficient design and control of continuous-variable quantum states.

3. Cross-Cutting Themes and Perspectives

Although the contributions to this Special Issue address different classes of special functions and applications, several recurring themes emerge.

3.1. Analytic Representations and Complex-Analytic Structure

Complex analysis and analytic representations play a prominent role. The analytic continuation of $F_2$ relies on continued-fraction representations and explicit domain control in ${\mathbb{C}}^2$ [1]. In a different guise, complex-analytic structure enters the viscoelastic application through the branch structure and conjugate symmetry of the Lambert W function, which influences the associated spectral functions [2]. These works exemplify how analytic continuation and branch-cut properties can guide modeling and interpretation.

3.2. Transform Techniques for Stability and Computation

Integral transforms provide a bridge between analysis and applications. Transform-based stability theory is central to the Ulam–Hyers analysis of linear differential equations [4]. Transform methods also appear in the computational setting: series expansions based on Bell polynomials yield a constructive route to Laplace transforms of nested functions [6]. Together, these papers highlight how transform-domain reasoning can simplify proofs, enable estimates, and suggest efficient computational procedures.

3.3. Special Polynomials and Discrete Structures

Special polynomials and combinatorial sequences act as unifying tools across seemingly disparate settings. Orthogonal polynomials (shifted Vieta–Lucas) are used as approximation bases for multi-space fractional PDEs [3]. Operator constructions based on monomiality generate new Hermite-type families with prescribed algebraic properties [5]. On the discrete side, higher-level poly-Cauchy numbers extend classical special-number hierarchies and connect to generating-function techniques [7]. Bell polynomials, in turn, provide explicit combinatorial coefficients in transform expansions [6]. These developments demonstrate that polynomial and combinatorial calculi remain central to both analysis and computation.

3.4. Symmetry as an Organizing Principle

This Special Issue is published in Symmetry, and symmetry is indeed a recurring organizing principle. Domain symmetry underlies the analytic-continuation results for multivariate hypergeometric functions [1]. Symmetry also appears in algebraic operator frameworks [5] and in the conjugation properties that control the spectral representations of Lambert W [2]. In quantum information, symplectic symmetry structures the classification of Gaussian states and connects abstract invariants to practical optical implementations [8,9]. The collection thus illustrates how symmetry can unify methods across analysis, algebra, and physical modeling.

4. Conclusions

The nine papers in this Special Issue collectively demonstrate that the theory and applications of special functions remain a vibrant and evolving field. From the extension of classical hypergeometric functions and the exploration of new polynomial families to innovative numerical methods and quantum state engineering, the breadth of topics covered is testimony to the rich interplay between abstract mathematical concepts and practical problems. Theory and Applications of Special Functions II continues the effort to bridge pure mathematics with real-world applications, highlighting how symmetry principles and special functions can lead to novel solutions across disciplines. It is our hope that these contributions will inspire further research and collaboration, and that the advancements reported here will stimulate new approaches to the use of symmetry and special functions in scientific research.