Search Results (408)

Closed-form analytic inverses allow explicit tracking of parameter effects, facilitate interpretation of experimental signals, and support solving inverse problems. Here, we derive a rigorous closed-form expression for the inverse Laplace transform of a class of shifted quasi-rational spectral functions with a square-root radical and a power-law decaying factor. These functions appear in coupled diffusion processes in physics and in the analysis of electromagnetic signal propagation through electrically cascaded networks, signal processing, and related areas. The transform is expressed as a finite sum of three generalized hypergeometric functions—two Kummer functions and one five-parameter Kampé de Fériet function—each multiplied by a monomial depending on the decay parameter. The validity of the result is confirmed by direct Laplace transformation, which recovers the original spectral function. Several known inverse transforms appear as limiting cases, illustrating the generality of the solution. Additionally, reduction formulas for a subclass of Kampé de Fériet functions demonstrate how the general solution encompasses previously known results and highlight the generality of the method. Full article

The present study investigates the generalized dispersion of a solute in an incompressible MHD flow via a rectangular channel with injectable or suctioned walls. The mathematical model of dispersion suggests a distinct type of mass flux expressed as a fractional partial differential equation based on the time-fractional Caputo derivative. The mass flow in the model under investigation is determined by both the concentration gradient and its historical evolution. A constant external magnetic field is provided transverse to the flow direction. The analysis and discussion of the analytical solution for the advection velocity are performed in relation to the Hartmann number and the suction/injection Reynolds number. To determine the solute concentration in space and time, the unstable fractional convection–diffusion equation is analytically solved. The polynomial in the geographic variable y that has coefficients that depend on the spatial variable x and the time t is the analytical solution of the concentration. The effects of the fractional order of the Caputo derivative, Reynolds number, Hartmann number, and Peclet number on the advection–diffusion process are examined using numerical simulations of the analytical solution of the solute concentration. Full article

In the theory of mixed-type equations, there are many works in bounded domains with smooth boundaries bounded by a normal curve for first- and second-kind mixed-type equations. In this paper, for a second-kind mixed-type equation in an unbounded domain whose elliptic part is a horizontal half-strip, a Bitsadze–Samarskii-type problem is investigated. The uniqueness of the solution is proved using the extremum principle, and the existence of the solution is proved by the Green’s function method and the integral equations method. When constructing the Green’s function, the properties of Bessel functions of the second kind with imaginary argument and the properties of the Gauss hypergeometric function are widely used. Visualization of the solution to the Bitsadze–Samarskii-type problem is performed, confirming its correctness from both mathematical and physical points of view. Full article

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. Full article

Mock theta functions, introduced by Ramanujan in his last letter to Hardy, play a significant role in q-series theory and have natural connections to fractional q-calculus. In this paper, we study bilateral hypergeometric series of the form ${}_2\psi _2$= ${\sum }_{n=-\infty }^{\infty }{\displaystyle \frac{{(a,b;q)}_n}{{(c,d;q)}_n}}{z}^n,$ where ${(a;q)}_n$ denotes the q-shifted factorial. Using Slater’s three-term transformation formula for bilateral ${}_2\psi _2$ series, we derive new identities for Ramanujan’s mock theta functions of orders 2, 3, 6, and 8. These transformations reveal previously unknown relationships between different q-series representations and extend the classical theory of mock theta functions within the framework of q-special functions. Full article

This paper aims to obtain the ${}_{\xi }\mathsf{\Psi }_{\eta }$-extended Whittaker function and its integral representations. This function is defined by using the ${}_{\xi }\mathsf{\Psi }_{\eta }$-confluent hypergeometric function, which was recently extended in terms of the Fox–Wright function. Furthermore, we discuss properties including a transformation formula, integral transforms (Laplace–Mellin and Hankel transforms), and a differential formula. Our results provide a unified framework for several known generalizations of the Whittaker function and highlight potential applications in applied mathematics and theoretical physics. Full article

In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By employing Binet-type formulas, recurrence relations, Chebyshev polynomial connections, and classical transformation properties of Gauss hypergeometric functions, we derive several explicit and alternative representations for the Pell and Jacobsthal numbers. These representations unify known identities, yield new closed-form expressions, and reveal deeper structural parallels between hypergeometric functions and linear recurrence sequences. The results demonstrate that hypergeometric functions provide a systematic and versatile analytical tool for studying special number sequences beyond the Fibonacci case, and they suggest potential extensions to broader families such as Horadam-type sequences and their generalizations. Full article

A transformation method was applied to the biconfluent (BHE), doubly confluent (DHE), and triconfluent (THE) Heun equations to generate and classify exactly solvable quantum mechanical potentials derived from them. With this, the range of potentials solvable in terms of the confluent hypergeometric function can be extended. The resulting potentials contained five independently tunable terms and two terms originating from the Schwartzian derivative that depended only on the parameters of the $z\left(x\right)$ transformation function. The polynomial solutions of these potentials contain expansion coefficients obtained from three-term (BHE and DHE) and four-term (THE) recurrence relations. For the simplest $z\left(x\right)$ transformation functions, the Lemieux–Bose potentials have been recovered for the BHE and DHE. The coupling parameters of these potentials and also of five potentials derived from the THE have been expressed in terms of the parameters of the respective differential equations. The present scheme offers a general framework into which a number of earlier results can be integrated in a systematic way. These include special cases of potentials obtained from less general versions of the Heun-type equations and individual solvable potentials obtained from various methods that do not necessarily refer to the Heun-type equations considered here. Several potentials derived here were found to coincide with or reduce to potentials found earlier from the quasi-exactly solvable (QES) formalism. Based on their mathematical form, their physically relevant features (domain of definition, asymptotic behaviour, single- or multi-well structure) were discussed, and possible fields of applications were pointed out. Full article

This article defines an inverted Topp–Leone distribution. Several mathematical properties and maximum likelihood estimation of parameters of this distribution are considered. The shape of the distribution for different sets of parameters is discussed. Several mathematical properties such as the cumulative distribution function, mode, moment-generating function, survival function, hazard rate function, stress-strength reliability R, moments, Rényi entropy, Shannon entropy, Fisher information matrix, and partial ordering associated with this distribution, have been derived. Distributions of the sum and quotient of two independent inverted Topp–Leone variables have also been obtained. Full article

We derive an explicit analytic expression for the first quantum correction to the second virial coefficient of a d-dimensional fluid whose particles interact via the generalized Lennard-Jones $(2n,n)$ potential. By introducing an appropriate change of variable, the correction term is reduced to a single integral that can be evaluated in closed form in terms of parabolic cylinder or generalized Hermite functions. The resulting expression compactly incorporates both dimensionality and stiffness, providing direct access to the low- and high-temperature asymptotic regimes. In the special case of the standard Lennard-Jones fluid ($d=3$, $n=6$), the formula obtained is considerably more compact than previously reported representations based on hypergeometric functions. The knowledge of this correction allows us to determine the first quantum contribution to the Boyle temperature, whose dependence on dimensionality and stiffness is explicitly analyzed, and enables quantitative assessment of quantum effects in noble gases such as helium, neon, and argon. Moreover, the same methodology can be systematically extended to obtain higher-order quantum corrections. Full article

The paper considers the problem of the analytical extension of the ratios of generalized hypergeometric functions ${}_{3}F_2$ A new domain of analytic continuation for these ratios under certain conditions to parameters is established. In this case, the domain of analytic extension of the special function is the domain of convergence of its branched continued fraction expansion. This paper also provides an example of applying the obtained results to dilogarithm function. Full article

We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct, new summation formulas with finite sums involving the psi function and a recursive formula for Bateman’s G function are derived. Finally, all the results have been numerically checked with MATHEMATICA. Full article

In this study, we consider and analyze an inhomogeneous Whittaker-type differential equation of the form ${\displaystyle \frac{d^{2}y\left(x\right)}{d{x}^2}}+{\displaystyle \frac{1}{x}}{\displaystyle \frac{dy\left(x\right)}{dx}}-{\alpha }^2\left(x^2-{\beta }^2\right)y\left(x\right)=g\left(x\right)$, where $\alpha$ and $\beta$ are given parameters. We investigate the analytical structure of its solution through the application of the Whittaker integral representation. The analysis encompasses both initial value problems (IVPs) and boundary value problems (BVPs), wherein appropriate conditions are imposed within a unified analytical framework. Furthermore, a systematic methodology is developed for constructing explicit solutions within the framework of Whittaker function theory. This approach not only elucidates the functional behaviour of the solutions but also provides a foundation for extending the analysis to more general classes of second-order linear differential equations. Full article

In this study, we introduced and analyzed the Slash–Log–Logistic (SlaLL) distribution, a novel statistical model developed by applying the slash methodology to log–logistic and beta distributions. The SlaLL distribution is particularly suited for modeling datasets characterized by heavy tails and extreme values, frequently encountered in survival time analyses. We derived the mathematical representation of the distribution involving Gauss hypergeometric and beta functions, explicitly established the probability density function, cumulative distribution function, hazard rate function, and reliability function, and provided clear definitions of its moments. Through comprehensive simulation studies, the accuracy and robustness of maximum likelihood and Bayesian methods for parameter estimation were validated. Comparative empirical analyses demonstrated the SlaLL distribution’s superior fitting performance over well-known slash-based models, emphasizing its practical utility in accurately capturing the complexities of real-world survival time data. Full article

Cell–cell interactions play a pivotal role in maintaining tissue homeostasis and driving disease progression. Conventional Cell–cell interactions modeling approaches depend on ligand–receptor databases, which often fail to capture context-specific or newly emerging signaling mechanisms. To address this limitation, we propose a data-driven computational framework, data-driven cell–cell interaction inference (DD-CC-II), which employs a graph-based model using eigen-cells to represent cell groups. DD-CC-II uses eigen-cells (i.e., functional module within the cell population) to characterize cell groups and construct correlation coefficient networks to model between-group associations. Correlation coefficient networks between eigen-cells are constructed, and their statistical significance is evaluated via over-representation analysis and hypergeometric testing. Monte Carlo simulations demonstrate that DD-CC-II achieves superior performance in inferring CCIs compared with ligand–receptor-based methods. The application to whole-blood RNA-seq data from the Japan COVID-19 Task Force revealed severity stage-specific interaction patterns. Markers such as FOS, CXCL8, and HLA-A were associated with high severity, whereas IL1B, CD3D, and CCL5 were related to low severity. The systemic lupus erythematosus pathway emerged as a potential immune mechanism underlying disease severity. Overall, DD-CC-II provides a data-centric approach for mapping the cellular communication landscape, facilitating a better understanding of disease progression at the intercellular level. Full article