Tricomi Functions Research Articles

Borel transform and integral identities involving λ$$ \lambda $$‐Bessel and λ$$ \lambda $$‐Tricomi matrix functions

This paper focuses on integrals and Borel transforms concerning ‐Bessel and ‐Tricomi functions, employing an umbral perspective as a central approach. Differintegral techniques, presently utilized in calculus, offer a rich reserve of tools that can be utilized across a broad spectrum of problems encompassing the realm of special functions and beyond. Employing integral transforms akin to the Borel type and the related formalism proves to be a potent approach, facilitating a connection between umbral and operational methodologies. By integrating these perspectives, we establish a novel and effective method for deriving integrals of hybrid special functions and achieving the summation of their corresponding generating functions.

On Approximation Operators Involving Tricomi Function

On Approximation Operators Involving Tricomi Function

Two-variable q-Laguerre polynomials from the context of quasi-monomiality

We introduce the theory of two-variable q-Laguerre polynomials by means of a generating function involving zeroth-order q-Bessel Tricomi functions and we establish two-variable q-Laguerre polynomials from the context of quasi-monomiality. Then we obtain the operational representations and q-integrodifferential equations for two-variable q-Laguerre polynomials. Moreover, we introduce mth-order two-variable q-Laguerre polynomials, we deduce the q-partial differential equations and q-integrodifferential equations for mth-order two-variable q-Laguerre polynomials and we also examine the quasi-monomiality characteristics of these new q-special polynomials. Finally, we give some graphical representations of q-Laguerre polynomials.

Mittag-Leffler based Bessel and Tricomi functions via umbral approach

Mittag-Leffler based Bessel and Tricomi functions via umbral approach

Astigmatic transformation of a fractional-order edge dislocation

It is shown theoretically that an astigmatic transformation of an edge dislocation (straight line of zero intensity) of the ν-th order (ν=n+α is a real positive number, n is integer, 0<α<1 is the fractional part of the number) forms at twice the focal length from a cylindrical lens n optical elliptical vortices (screw dislocations) with a topological charge of –1, located on a straight line perpendicular to the edge dislocation. Coordinates of these points are zeros of the Tricomi function. At some distance from these vortices and on the same straight line, another additional vortex with a topological charge of –1 is also generated, which moves to the periphery if α decreases to zero, or approaches n vortices if α tends to 1. In addition, at the periphery in the beam cross-section, a countable number of optical vortices (intensity zeros) are formed, all with a topological charge of –1, which are located on diverging curved lines (such as hyperbolas) equidistant from a straight line on which the main n intensity zeros are located. These "accompanying" vortices approach the center of the beam, following the additional "passenger" vortex, if 0<α<0.5, or move to the periphery, leaving the "passenger" next to the main vortices, if 0.5<α<1. At α=0 and α=1, the "accompanying" vortices are situated at infinity. The topological charge of the entire beam at fractional ν is infinite. The numerical simulation confirms theoretical predictions.

An astigmatic transform of a fractional-order edge dislocation

In this work, it is theoretically and numerically demonstrated that an astigmatic transformation of a νth-order edge dislocation (shaped as a zero-intensity straight line) of a coherent light field—where ν =n + α is a real positive number, n is integer, and 0 <α <1 is fractional—produces n optical elliptic vortices (screw dislocations) with topological charge (TC) −1, which are arranged on a straight line perpendicular to the edge dislocation and found at Tricomi function zeros. We also reveal that at a distance from the said optical vortices (OV), an extra OV with charge −1 is born on the same straight line, which departs to the periphery with α tending to zero, or gets closer to the n OVs with α tending to 1. Additionally, we find that a countable number of OVs (intensity nulls) with charge −1 are produced at the field periphery and arranged on diverging hyperbolic curves equidistant from the straight line of the n main intensity nulls. These additional OVs, which we term as ‘escort’, either approach the beam center, accompanying the extra ‘companion’ OV if 0 <α <0.5, or depart to the periphery, whereas the ‘companion’ keeps close to the main OVs if 0.5 <α <1. At α =0 or α = 1, the ‘escort’ OVs are shown to be at infinity. At fractional ν, the TC of the whole optical beam is theoretically shown to be infinite. Numerical simulation results are in agreement with the theoretical findings.

A practical, effective calculation of gamma difference distributions with open data science tools

At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction and linear mixed models. The results of our numerical study, with emphasis on using open data science tools, demonstrate that our computational algorithm implemented in high-performance Python(with Numba) is exponentially fast, highly accurate and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our open tool outperformed the speed of the analytical computation based on Tricomi's function in CAS software (commercial Mathematica, open SageMath) by 1.5–2 orders. At the default precision of scientific numerical computational tools, it exceeded open SciPy, NumPy and commercial MATLAB 5–10 times. The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology, as well as in financial mathematics, and risk analysis.

A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation.

On a Solution of a Fractional Hyper-Bessel Differential Equation by Means of a Multi-Index Special Function

Abstract In this paper we introduce a new multiple-parameters (multi-index) extension of the Wright function that arises from an eigenvalue problem for a case of hyper-Bessel operator involving Caputo fractional derivatives. We show that by giving particular values to the parameters involved in this special function, this leads to some known special functions (as the classical Wright function, the α-Mittag-Leffler function, the Tricomi function, etc.) that on their turn appear as cases of the so-called multi-index Mittag-Leffler functions. As an application, we mention that this new generalization Wright function nis an isochronous solution of a nonlinear fractional partial differential equation.

Tricomi beams and nondiffracting sheet beams

We introduce a non-diffracting beam family as a new beam solution of the scalar Helmholtz equation in Cartesian coordinates whose complex amplitude is mainly described by the Tricomi functions and characterized by four parameters. Based on the proposed beam model, the on-axial, off-axial, and asymmetrical Bessel beams can be obtained as special cases through appropriate selection of the complex parameters. We show that both the intensity and phase distributions can be tailored in a controlled manner through modulation of the angular spectra in the frequency space, enabling the transverse intensity pattern to be varied from a concentric ring structure, through a concentric arc configuration, to a single sheet pattern with different orientations and central positions.

Accessible solitons in three-dimensional parabolic cylindrical coordinates

A class of self-similar beams, named three-dimensional (3D) spatiotemporal parabolic accessible solitons, are introduced in the 3D highly nonlocal nonlinear media. We obtain exact solutions of the 3D spatiotemporal linear Schrödinger equation in parabolic cylindrical coordinates by using the method of separation of variables. The 3D localized structures are constructed with the help of the confluent hypergeometric Tricomi functions and the Hermite polynomials. Based on such an exact solution, we graphically display three different types of 3D beams: the Gaussian solitons, the ring necklace solitons, and the parabolic solitons, by choosing different mode parameters. We also perform direct numerical simulation to discuss the stability of local solutions. The procedure we follow provides a new method for the manipulation of spatiotemporal solitons.

Solving a nonlinear variation of the heat equation: self-similar solutions of the second kind and other results

This paper studies regular self-similar solutions of the following diffusion equation $$\begin{aligned} u_{t}+\gamma \vert u_{t} \vert =\Delta u\quad \text {in}\ \mathbb {R}^{N}\times ]0,\infty [, \end{aligned}$$ where $$-1<\gamma <1$$ . The analysis is focused on radial symmetric solutions $$u(x,t)=t^{-\alpha /2}f(\eta )$$ with $$\alpha >0$$ and $$\eta =\Vert x\Vert /\sqrt{t}$$ . Closed representation is obtained in terms of confluent hypergeometric functions. Employing specific properties of these special functions, oscillatory and symptotic aspects of f are obtained. It is demonstrated that such features are governed by increasing and unbounded sequences of exponents $$\alpha _{0}<\alpha _{1}<\cdots $$ , as in other diffusion equations. These exponents are determined by solving a system of transcendental equations related to specific roots of Kummer and Tricomi functions. As these cannot be determined using dimensional analysis, it is concluded that they are anomalous. For each exponent $$\alpha _{k}$$ , linear approximation when $$\gamma $$ is close to zero is also presented. Finally, relationships with previous results as well as an extension to other fully nonlinear parabolic equations are discussed.

АНАЛІЗ ЗАГАЛЬНОГО РОЗВ’ЯЗКУ ОДНОГО ЛІНІЙНОГО ОДНОРІДНОГО ДИФЕРЕНЦІАЛЬНОГО РІВНЯННЯ З ПРАВИЛЬНОЮ ОСОБЛИВОЮ ТОЧКОЮ

У роботі виявлено, що обране для дослідження лінійне однорідне диференціальне рівняння другого порядку, яке має правильну особливу точку, за допомогою перетворення Куммера-Ліувілля зводиться до виродженого гіпергеометричного рівняння, яке подане в формі Уіттекера. Встановлені співвідношення між коефіцієнтами заданого рівняння, виродженого гіпергеометричного рівняння у загальному вигляді та у формі Уіттекера. На основі цих співвідношень показано, що досліджуване рівняння має лише один незалежний фундаментальний розв’язок у вигляді виродженої гіпергеометричної функції. Другий фундаментальний розв’язок може бути знайдений або за допомогою метода Фробеніуса у вигляді узагальненого степеневого ряду, або виражений через функцію Трікомі. В останньому випадку загальний розв’язок досліджуваного рівняння є лінійною комбінацією функцій Уіттекера.

Inequalities for Tricomi functions

Inequalities for Tricomi functions

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.

Certain results concerning operators on a Lie algebra and double index sequences

Abstract In this paper, a result for double index sequences is established by using a quadratic combination of four operators defined on a Lie algebra of endomorphisms of a vector space. This result is used to give new proofs of some properties of the Bessel and Tricomi functions of two indices. Further, this result is extended for multi-index sequences.

New classes of Hermite–Tricomi functions and generating relations

Abstract. In this paper, new and general classes of multi-variable Hermite–Tricomi functions are introduced and some properties of these functions are established. Further, generating relations for Laguerre–Hermite polynomials involving various forms of Hermite–Tricomi functions are derived.

Generating relations of multi-variable Tricomi functions of two indices using Lie algebra representation

This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive certain generating relations involving 2-indices 5-variables 5-parameters Tricomi functions (2I5V5PTF) by using a Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.

The Wigner distribution functions of coherent and partially coherent Bessel—Gaussian beams

Based on the integral representation of the Bessel functions and the generating function of the Tricomi function, an analytical expression of the Wigner distribution function (WDF) for a coherent or partially coherent Bessel—Gaussian beam is presented. The reduced two-dimensional WDFs are also demonstrated graphically, which reveals the dependence of the reduced WDFs on the beam parameters.

On Tricomi and Hermite-Tricomi Matrix Functions of Complex Variable

In this paper, Tricomi and Hermite-Tricomi matrix functions are introduced starting from the Hermite matrix polynomials. The convergence, radius of regularity, integral form, generating matrix functions, matrix recurrence relations satisfied by these Tricomi matrix functions are derived. Finally, the generating matrix functions, matrix recurrence relations, addition theorems for the Hermite-Tricomi matrix functions are given and matrix differential equations satisfied by them are presented.