Search Results (13)

We analyze the spherically symmetric complex diffusion and special type of the complex reaction–diffusion equations. These equations are form invariant to the free Schrödinger equations and to the Schrödinger equations with power-law space-dependent potentials. Our new type of solutions are important because we found a new realm of solutions which lie between the solutions of the classical regular diffusion equation and the usual quantum mechanical solutions of the Schrödinger equation. As the solution method, we applied the the self-similar Ansatz, which reduces the original partial differential equation (PDE) to an ordinary differential equation (ODE) which can be solved analytically. The self-similar Ansatz couples the spatial and temporal variables together instead of the usual separation which has to be used in ordinary quantum mechanics for time-independent Hamiltonian. For the complex diffusion equation—without any additional source term—the solutions are the Kummer’s M and Kummer’s U functions. For some parameter values we found $L^2$ integrability, as in the Cartesian case. We interpret that this property can be a “quantum mechanical heritage” and can be a far relation to ordinary quantum mechanics. Therefore, in this sense, our solutions might have quantum mechanical interest in the future. For the complex reaction–diffusion-type equation we derived the Whittaker M and Whittaker W functions as solutions. These solutions have no $L^2$ integrability at all. All derived solutions have complex quadratic arguments. These kind of analytic solutions are new and cannot be found in the existing scientific literature. Finally, the role of the complex angular momentum was investigated as well. 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

The simultaneous detection of multiple cancer biomarkers using microfluidic multiplexed immunosensors is gaining significant interest in the field of Point-of-Care diagnostics. This study highlights integrating surface-enhanced infrared Fourier transform (SE-FTIR) with a plasmonic-active nanostructure thin film (PANTF) on a printed circuit board (PCB), housed within a microfluidic device for rapid, non-destructive detection of breast cancer (BC). Detection uses monoclonal antibody (mAb)-functionalized gold nanourchins (GNUs) on dual sensing regions. A total of 12 serum samples (24 data points) were tested for HER-II and CA 15-3. The system demonstrated a SE-FTIR enhancement factor (EF) of ~0.18 × 10⁵ using Rhodamine 6G (R6G). Calibration with HER-II (1–100 ng/mL) and CA 15-3 (10–100 U/mL) showed linear responses (R² = 0.8 and 0.76, respectively). Measurements of unknowns were performed at 1 µL/min over 68 min, with 43 min for biomarker interaction. SE-FTIR spectra were recorded at active zones and analyzed using SpectraView (SV), a custom Python 3.12-based tool. Data preprocessing included filtering (SciPy’s filtfilt) and baseline correction using the Improved Asymmetric Least Squares (IASLS) algorithm (pybaselines.Whittaker). Fourier cross-correlation (FCC) showed stronger signal consistency for HER-II. Partial Least Squares (PLS) regression, a dimensionality reduction technique, enabled clear discrimination between the samples and types, with classification accuracy reaching 1.0. Cancer staging based on these biomarkers yielded an overall accuracy of 0.54, indicating that classification regardless of biomarker type. Further studies involving larger and more diverse sample sets are critical before any definitive conclusions can be drawn. 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

The second virial coefficient (SVC) of the Lennard-Jones fluid is a cornerstone of molecular theory, yet its calculation has traditionally relied on the complex integration of the pair potential. This work introduces a fundamentally different approach by reformulating the problem in terms of ordinary differential equations (ODEs). For the classical component of the SVC, we generalize the confluent hypergeometric and Weber–Hermite equations. For the first quantum correction, we present entirely new ODEs and their corresponding exact-analytical solutions. The most striking result of this framework is the discovery that these ODEs can be transformed into Schrödinger-like equations. The classical term corresponds to a harmonic oscillator, while the quantum correction includes additional inverse-power potential terms. This formulation not only provides a versatile method for expressing the virial coefficient through a linear combination of functions (including Kummer, Weber, and Whittaker functions) but also reveals a profound and previously unknown mathematical structure underlying a classical thermodynamic property. Full article

In this study, a semi-analytical solution to the inhomogeneous Whittaker equation is developed for both initial and boundary value problems. A new class of special integral functions ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$, along with their derivatives, is introduced to facilitate the construction of the solution. The analytical properties of ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ are rigorously investigated, and explicit closed-form expressions for ${}^f\mathrm{Zi}_{\kappa ,\mu }\left(x\right)$ and its derivatives are derived in terms of Whittaker functions ${\mathrm{M}}_{\kappa ,\mu }\left(z\right)$ and ${\mathrm{W}}_{\kappa ,\mu }\left(z\right)$, confluent hypergeometric functions, and other special functions including Bessel functions, modified Bessel functions, and the incomplete gamma functions, along with their respective derivatives. These expressions are obtained for specific parameter values using symbolic computation in Maple. The results contribute to the broader analytical framework for solving inhomogeneous linear differential equations with applications in engineering, mathematical physics, and biological modeling. Full article

We present a novel proof, using group theory, for a Meijer transform formula. This proof reveals the formula as a specific case of a broader generalized result. The generalization is achieved through a linear operator that intertwines two representations of the connected component of the identity of the group $\mathrm{SO}(2,2)$. Using this same approach, we derive a formula for the sum of three double integral transforms, where the kernels are represented by Bessel functions. It is particularly noteworthy that the group $\mathrm{SO}(2,2)$ is connected to symmetry in several significant ways, especially in mathematical physics and geometry. Full article

We propose a method for generating three-dimensional light fields with given intensity and phase distributions using purely phase transmission functions. The method is based on a generalization of the well-known approach to the design of diffractive optical elements that focus an incident laser beam into an array of light spots in space. To calculate purely phase transmission functions, we use amplitude encoding, which made it possible to implement the designed elements using a single spatial light modulator. The generation of light beams in the form of rings, spirals, Lissajous figures, and multi-petal “rose” distributions uniformly elongated along the optical axis in the required segment is demonstrated. It is also possible to control the three-dimensional structure of the intensity and phase of the shaped light fields along the propagation axis. The experimentally generated intensity distributions are in good agreement with the numerically obtained results and show high potential for the application of the proposed method in laser manipulation with nano- and microparticles, as well as in laser material processing. Full article

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$. In this second part, first derivatives with respect to the parameters of the Whittaker function ${\mathrm{W}}_{\kappa ,\mu }\left(x\right)$ are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, it is possible to obtain these parameter derivatives in terms of infinite integrals, with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions), from the integral representation of ${\mathrm{W}}_{\kappa ,\mu }\left(x\right)$. These infinite sums and integrals can be expressed in closed form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function ${\mathrm{wi}}_{\kappa ,\mu }\left(x\right)$ and its derivative with respect to $\kappa$, as well as some reduction formulas for the integral Whittaker functions ${\mathrm{Wi}}_{\kappa ,\mu }\left(x\right)$ and ${\mathrm{wi}}_{\kappa ,\mu }\left(x\right)$, are calculated. Full article

In this paper, first derivatives of the Whittaker function ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions ${\mathrm{M}}_{\kappa ,\mu }\left(x\right)$ and integral Whittaker functions ${\mathrm{Mi}}_{\kappa ,\mu }\left(x\right)$ and ${\mathrm{mi}}_{\kappa ,\mu }\left(x\right)$ are calculated. Full article

In this paper, we shall establish a hierarchy of functional equations (as a G-function hierarchy) by unifying zeta-functions that satisfy the Hecke functional equation and those corresponding to Maass forms in the framework of the ramified functional equation with (essentially) two gamma factors through the Fourier–Whittaker expansion. This unifies the theory of Epstein zeta-functions and zeta-functions associated to Maass forms and in a sense gives a method of construction of Maass forms. In the long term, this is a remote consequence of generalizing to an arithmetic progression through perturbed Dirichlet series. Full article

The Voronoĭ summation formula is known to be equivalent to the functional equation for the square of the Riemann zeta function in case the function in question is the Mellin tranform of a suitable function. There are some other famous summation formulas which are treated as independent of the modular relation. In this paper, we shall establish a far-reaching principle which furnishes the following. Given a zeta function $Z\left(s\right)$ satisfying a suitable functional equation, one can generalize it to $Z_f\left(s\right)$ in the form of an integral involving the Mellin transform $F\left(s\right)$ of a certain suitable function $f\left(x\right)$ and process it further as ${\tilde{Z}}_f\left(s\right)$. Under the condition that $F\left(s\right)$ is expressed as an integral, and the order of two integrals is interchangeable, one can obtain a closed form for ${\tilde{Z}}_f\left(s\right)$. Ample examples are given: the Lipschitz summation formula, Koshlyakov’s generalized Dedekind zeta function and the Plana summation formula. In the final section, we shall elucidate Hamburger’s results in light of RHBM correspondence (i.e., through Fourier–Whittaker expansion). Full article

Integral Mittag-Leffler, Whittaker and Wright functions with integrands similar to those which already exist in mathematical literature are introduced for the first time. For particular values of parameters, they can be presented in closed-form. In most reported cases, these new integral functions are expressed as generalized hypergeometric functions but also in terms of elementary and special functions. The behavior of some of the new integral functions is presented in graphical form. By using the MATHEMATICA program to obtain infinite sums that define the Mittag-Leffler, Whittaker, and Wright functions and also their corresponding integral functions, these functions and many new Laplace transforms of them are also reported in the Appendices for integral and fractional values of parameters. Full article