Bessel-type Functions Research Articles

On some $q$-Bessel type continuous wavelet transform

In this paper we continue as in \cite{Rezguietal} to exploit the modified variants of Bessel function in the framework of $q$-theory to construct wavelet operators. A generalized $q$-Bessel type function has been introduced leading to an associated mother wavelet which in turns has induced a continuous wavelet transform. Finally, Plancherel/Parceval type relations have been proved. Such variants of wavelets permit in some sense to approximate solutions of ODEs and PDEs by transforming them to recurrent sequences.

Quaternionic Convolutional Neural Networks with Trainable Bessel Activation Functions

Quaternionic convolutional neural networks (QCNN) possess the ability to capture both external dependencies between neighboring features and internal latent dependencies within features of an input vector. In this study, we employ QCNN with activation functions based on Bessel-type functions with trainable parameters, for performing classification tasks. Our experimental results demonstrate that this activation function outperforms the traditional ReLU activation function. Throughout our simulations, we explore various network architectures. The use of activation functions with trainable parameters offers several advantages, including enhanced flexibility, adaptability, improved learning, customized model behavior, and automatic feature extraction.

Bicomplex Neural Networks with Hypergeometric Activation Functions

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.

Taylor Series for the Mittag–Leffler Functions and Their Multi-Index Analogues

It has been obtained that the n-th derivative of the 2-parametric Mittag–Leffler function is a 3-parametric Mittag–Leffler function, with exactness to a constant. Following the analogy, the author later obtained the n-th derivative of the 2m-parametric multi-index Mittag–Leffler function. It turns out that this is expressed via the 3m-parametric Mittag–Leffler function. In this paper, upper estimates of the remainder terms of these derivatives are found, depending on n. Some asymptotics are also found for “large” values of the parameters. Further, the Taylor series of the 2 and 2m-parametric Mittag–Leffler functions around a given point are obtained. Their coefficients are expressed through the values of the corresponding n-th order derivatives at this point. The convergence of the series to the represented Mittag–Leffler functions is justified. Finally, the Bessel-type functions are discussed as special cases of the multi-index (2m-parametric) Mittag–Leffler functions. Their Taylor series are derived from the general case as corollaries, as well.

Comparative performance of different hyperbolic cosine functions and generalized B functions basis sets for atomic systems

Recently, we reported a new set of Bessel type functions, which include the radial part of generalized Bessel functions for LCAO calculations of atomic systems. In this study, to achieve further improvement of the performance of generalized Bessel type basis sets in the Hartree–Fock-Roothaan calculations, different hyperbolic cosine functions inserted into the radial part of those generalized Bessel functions. For this purpose, three different generalized hyperbolic cosine functions have been used to construct the generalized Bessel type hyperbolic cosine basis sets. The accuracies of generalized Bessel type hyperbolic cosine functions within the minimal basis sets approach are compared to show their superiority to conventional approaches those in the literature. The performance of the presented basis functions is also compared to the numerical Hartree–Fock results. Our virial ratios are in good agreement to within 8-digits of the −2. It is shown that the results obtained by the new basis sets surpass the quality and accuracy of existing Bessel type basis sets.

Humbert Polynomials and Functions in Terms of Hemite Polynomials

By starting from the standard definitions of the incomplete two-variable Hermite polynomials, we propose non-trivial generalizations and we show some applications to the Bessel-type functions as the Humbert functions. We also present a generalization of the Laguerre polynomials in the same context of the incomplete-type and we use these to obtain relevant operational techniques for the Humbert-type functions.

Self-Similar Functional Circuit Models of Arteries and Deterministic Fractal Operators: Theoretical Revelation for Biomimetic Materials.

In this paper, the self-similar functional circuit models of arteries are proposed for bioinspired hemodynamic materials design. Based on the mechanical-electrical analogous method, the circuit model can be utilized to mimic the blood flow of arteries. The theoretical mechanism to quantitatively simulate realistic blood flow is developed by establishing a fractal circuit network with an infinite number of electrical components. We have found that the fractal admittance operator obtained from the minimum repeating unit of the fractal circuit can simply and directly determine the blood-flow regulation mechanism. Furthermore, according to the operator algebra, the fractal admittance operator on the aorta can be represented by Gaussian-type convolution kernel function. Similarly, the arteriolar operator can be described by Bessel-type function. Moreover, by the self-similar assembly pattern of the proposed model, biomimetic materials which contain self-similar circuits can be designed to mimic physiological or pathological states of blood flow. Studies show that the self-similar functional circuit model can efficiently describe the blood flow and provide an available and convenient structural theoretical revelation for the preparation of in vitro hemodynamic bionic materials.

On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space

In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.

On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions

ABSTRACTIn the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to the Laguerre weight and an additional point mass N>0 at the origin, these polynomials satisfy, for any , a linear differential equation of order . In the present paper we establish a new elementary representation of the corresponding 'Laguerre-type' differential operator and show its symmetry with respect to the underlying weighted scalar product. Furthermore, we discuss various other representations of the operator, mainly given in factorized form, and show their equivalence. Finally, by applying a limiting process to the Laguerre-type equation, we deduce new elementary representations for the higher-order differential equation satisfied by the Bessel-type functions on the positive half-line.

Certain inequalities for the modified Bessel-type function

We establish some new inequalities for the modified Bessel-type function lambda _{nu ,sigma }^{(beta )} (x ) studied by Glaeske et al. [in J. Comput. Appl. Math. 118(1–2):151–168, 2000] as the kernel of an integral transformation that modifies Krätzel’s integral transformation. The inequalities obtained are closely related to the generalized Hurwitz–Lerch zeta function and complementary incomplete gamma function. We also deduce some useful inequalities for the modified Bessel function of the second kind K_{nu } (x ) and Mills’ ratio mathsf{M} (x ) as worthwhile applications of our main results.

On the Radial Eigenmode Structure of Drift Wave Instability with Inhomogeneous Damping in Cylindrical Plasmas

Plasma flows can be driven by turbulent stresses from excited modes in magnetized plasmas. Our recent numerical simulation of resistive drift wave turbulence in a linear device has shown that the radial inhomogeneity of the neutral density affects azimuthal flow generation by changing the phase structure of the most unstable eigenmodes. Eigenmode analyses show that the mode structure has a complex Bessel-type function shape in the central region of the plasma, and the imaginary part arises from the radial inhomogeneity of the damping term caused by ion-neutral collisions. The amplitude of turbulent stress is proportional to the inhomogeneity under a marginally stable condition. Global structural formation is an important factor for determining the plasma turbulent state, and this result clearly shows that several kinds of radial background distributions, the plasma and neutral densities in this case, can influence the global structures.

On Classical Multiplier Sequences

We provide the detailed proofs of two recent claims of Csordas and Forgács asserting that two particular Bessel-type functions generate classical multiplier sequences whose generic terms are Cauchy-products of Laguerre polynomials and hypergeometric functions, respectively. The way these sequences are generated involves functions from the Laguerre-Pólya class. We address the question whether these functions are unique in any way as generators of a given classical multiplier sequence.

Generalizations on Humbert polynomials and functions

AbstractBy starting from the standard definitions of the incomplete two-variable Hermite polynomials, we propose non-trivial generalizations and we show some applications to the Bessel-type functions as the Humbert functions. We present some relevant relations linking the Bessel-type functions, the Humbert functions and the generalized Hermite polynomials. We also present a generalization of the Laguerre polynomials in the same context of the incomplete-type and we see some useful operational relations involving special functions as the Tricomi functions; we use this family of Laguerre polynomials to introduce some operational techniques for the Humbert-type functions. Finally, we note as the formalism discussed could be exploited to generalize some distribution as for example the Poisson type.

On Schr\xf6dinger Operators with Inverse Square Potentials on the Half-Line

The paper is devoted to operators given formally by the expression -∂x2+α-141x2.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} -\\partial _x^2+\\left( \\alpha -\\frac{1}{4}\\right) \\frac{1}{x^{2}}. \\end{aligned}$$\\end{document}This expression is homogeneous of degree minus 2. However, when we try to realize it as a self-adjoint operator for real alpha , or closed operator for complex alpha , we find that this homogeneity can be broken. This leads to a definition of two holomorphic families of closed operators on L^2({mathbb {R}}_+), which we denote H_{m,kappa } and H_0^nu , with m^2=alpha , -1<mathrm{Re}(m)<1, and where kappa ,nu in {mathbb {C}}cup {infty } specify the boundary condition at 0. We study these operators using their explicit solvability in terms of Bessel-type functions and the Gamma function. In particular, we show that their point spectrum has a curious shape: a string of eigenvalues on a piece of a spiral. Their continuous spectrum is always [0,infty [. Restricted to their continuous spectrum, we diagonalize these operators using a generalization of the Hankel transformation. We also study their scattering theory. These operators are usually non-self-adjoint. Nevertheless, it is possible to use concepts typical for the self-adjoint case to study them. Let us also stress that -1<mathrm{Re}(m)<1 is the maximal region of parameters for which the operators H_{m,kappa } can be defined within the framework of the Hilbert space L^2({mathbb {R}}_+).

Multiplier sequences, classes of generalized Bessel functions and open problems

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier sequences interpolated by functions which are not entire and sums, averages and parametrized families of multiplier sequences. The main results include (i) the development of a ‘logarithmic’ multiplier sequence and (ii) several integral representations of a generalized Bessel-type function utilizing some ideas of G.H. Hardy and L.V. Ostrovskii. The explorations and analysis, augmented throughout the paper by a plethora of examples, led to a number of conjectures and intriguing open problems.

On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field

Because of the disconnectedness of a non-Archimedean ordered field in the topology induced by the order, it is possible to have non-constant functions with zero derivatives everywhere. In fact the solution space of the differential equation y′ = 0 is infinite dimensional. In this paper, we give sufficient conditions for a function on an open subset of the Levi-Civita field to have zero derivative everywhere and we use the nonconstant zero-derivative functions to obtain non-analytic solutions of systems of linear ordinary differential equations with analytic coefficients. Then we use the results to introduce Bessel-type special functions on the Levi-Civita field and to study some of their properties.

The Umbral operator and the integration involving generalized Bessel-type functions

Abstract The main purpose of this paper is to introduce a class of new integrals involving generalized Bessel functions and generalized Struve functions by using operational method and umbral formalization of Ramanujan master theorem. Their connections with trigonometric functions with several distinct complex arguments are also presented.

Coupling coefficients for tensor product representations of quantum SU(2)

We study tensor products of infinite dimensional representations (not corepresentations) of the $\mathrm{SU}(2)$ quantum group. Eigenvectors of certain self-adjoint elements are obtained, and coupling coefficients between different eigenvectors are computed. The coupling coefficients can be considered as $q$-analogs of Bessel functions. As a results we obtain several $q$-integral identities involving $q$-hypergeometric orthogonal polynomials and $q$-Bessel-type functions.

Statistical Characterization and Computationally Efficient Modeling of a Class of Underwater Acoustic Communication Channels

Underwater acoustic channel models provide a tool for predicting the performance of communication systems before deployment, and are thus essential for system design. In this paper, we offer a statistical channel model which incorporates physical laws of acoustic propagation (frequency-dependent attenuation, bottom/surface reflections), as well as the effects of inevitable random local displacements. Specifically, we focus on random displacements on two scales: those that involve distances on the order of a few wavelengths, to which we refer as small-scale effects, and those that involve many wavelengths, to which we refer as large-scale effects. Small-scale effects include scattering and motion-induced Doppler shifting, and are responsible for fast variations of the instantaneous channel response, while large-scale effects describe the location uncertainty and changing environmental conditions, and affect the locally averaged received power. We model each propagation path by a large-scale gain and micromultipath components that cumulatively result in a complex Gaussian distortion. Time- and frequency-correlation properties of the path coefficients are assessed analytically, leading to a computationally efficient model for numerical channel simulation. Random motion of the surface and transmitter/receiver displacements introduce additional variation whose temporal correlation is described by Bessel-type functions. The total energy, or the gain contained in the channel, averaged over small scale, is modeled as log-normally distributed. The models are validated using real data obtained from four experiments. Specifically, experimental data are used to assess the distribution and the autocorrelation functions of the large-scale transmission loss and the short-term path gains. While the former indicates a log-normal distribution with an exponentially decaying autocorrelation, the latter indicates a conditional Ricean distribution with Bessel-type autocorrelation.

On the characteristic function for asymmetric Student [formula omitted] distributions

Following up on the work of Nadarajah and Teimouri [Nadarajah, S., Teimouri, M., 2012. On the characteristic function for asymmetric exponential power distributions. Econometric Reviews 31, 475–481], we derive here, for the first time, explicit closed-form expressions for the characteristic function of the asymmetric Student t distribution. The expressions involve hypergeometric and Bessel type functions.