Tricomi Confluent Hypergeometric Functions Research Articles

Generation of hyper-bunched light by single Gaussian and non-Gaussian scattering processes

We derive theoretically that hyper-bunched light with a central normalized second-order correlation coefficient of six can be realized by a single Gaussian scattering process of parametric down conversion (PDC) light with a central normalized second-order correlation coefficient of three. The Gaussian scattering process is realized by a rotating ground-glass diffuser. We show that the photon counting probability distribution in this case obeys a Tricomi confluent hypergeometric function U[1+n,3/2,1/⟨n⟩] dependence. Furthermore, we also study non-Gaussian light-scattering probabilities that together with the different impinging light statistics give rise to new photon statistics accompanied by a variety of new values of the second-order correlation coefficient of the scattered light. These theoretical calculations suggest experiments using twin photons from a PDC process and characterizing their photon statistics properties before and after the scattering at the rotating diffuser. These investigations contribute to a more comprehensive understanding of the scattering process, the generated light, and new applications.

Hitting times for sticky skew CIR process

In this paper, we consider an extended skew CIR processes with sticky points, which is referred to as the sticky skew CIR process. We first calculate the infinitesimal generator and its domain. To explore its trajectory properties, we compute the Laplace transforms and the expectations of first hitting times over a constant boundary. The solutions of Laplace transforms are expressed in terms of Tricomi and Kummer confluent hypergeometric functions.

Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions

<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

Some inverse Laplace transforms that contain the Marcum Q function and an expanded property of the Marcum Q function

ABSTRACTThe Laplace transforms of the transition probability density and distribution functions of the Feller process contain products of a Kummer and a Tricomi confluent hypergeometric function. The intricacies caused by the singularity at 0 of the Feller process imply that ultimately seven new inverse Laplace transforms can be derived of which four contain the Marcum Q function. The results of this paper together with a scarcely used link between the Marcum and Nuttall Q functions also provide two alternative proofs for an existing identity involving two Marcum Q functions with reversed arguments. The paper also expands the existing expression for the Marcum Q function with identical arguments and order 1. In particular, the new formula applies to all integer and fractional values of the order and is expressed in terms of the generalized hypergeometric function.

Propagation of Scalar Fields in a Plane Symmetric Spacetime

The present article deals with solutions for a minimally coupled scalar field propagating in a static plane symmetric spacetime. The considered metric describes the curvature outside a massive infinity plate and exhibits an intrinsic naked singularity (a singular plane) that makes the accessible universe finite in extension. This solution can be interpreted as describing the spacetime of static domain walls. In this context, a first solution is given in terms of zero order Bessel functions of the first and second kind and presents a stationary pattern which is interpreted as a result of the reflection of the scalar waves at the singular plane. This is an evidence, at least for the massless scalar field, of an old interpretation given by Amundsen and Gron regarding the behaviour of test particles near the singularity. A second solution is obtained in the limit of a weak gravitational field which is valid only far from the singularity. In this limit, it was possible to find out an analytic solution for the scalar field in terms of the Kummer and Tricomi confluent hypergeometric functions.

Turán type inequalities for confluent hypergeometric functions of the second kind

In this paper we deduce some tight Turán type inequalities for Tricomi confluent hypergeometric functions of the second kind, which in some cases improve the existing results in the literature. We also give alternative proofs for some already established Turán type inequalities. Moreover, by using these Turán type inequalities, we deduce some new inequalities for Tricomi confluent hypergeometric functions of the second kind. The key tool in the proof of the Turán type inequalities is an integral representation for a quotient of Tricomi confluent hypergeometric functions, which arises in the study of the infinite divisibility of the Fisher-Snedecor F distribution.

Cooperative spectrum sharing MIMO systems with successive decoding

A spectrum sharing system with primary and secondary nodes, each equipped with an arbitrary number of antennas, is investigated. Particularly, the outage performance of an underlay cognitive system is analytically studied, in the case when the end-to-end (e2e) communication is established via an intermediate relay node. To better enhance the e2e communication, successive interference cancellation (SIC) is adopted, which compensates for both the transmission power constraint and the presence of interference from primary nodes. Both the relay and secondary receiver perform unordered SIC to successively decode the multiple streams, whereas the decode-and-forward relaying protocol is used for the e2e communication. New closed-form expressions for the e2e outage performance of each transmitted stream are derived in terms of finite sum series of the Tricomi confluent hypergeometric function. In addition, simplified yet tight approximations for the asymptotic outage performance are obtained. Useful engineering insights are manifested, such as the diversity order of the considered system and the impact of interference from the primary nodes in conjunction with the constrained transmission power of the secondary nodes.

On First Hitting Times for Skew CIR Processes

In this work, the first hitting times for skew CIR processes are investigated. We compute the Laplace transforms and the means of the first hitting times of some given levels. The solutions for Laplace transforms are in terms of Tricomi and Kummer confluent hypergeometric functions. We also exhibit the hitting time densities numerically at the end of this paper.

Inequalities for the One-Dimensional Analogous of the Coulomb Potential

In this paper our aim is to present some monotonicity and convexity properties for the one dimensional regularization of the Coulomb potential, which has applications in the study of atoms in magnetic fields and which is in fact a particular case of the Tricomi confluent hypergeometric function. Moreover, we present some Turan type inequalities for the function in the question and we deduce from these inequalities some new tight upper bounds for the Mills ratio of the standard normal distribution.

Asymptotic representations for hypergeometric-Bessel type function and fractional integrals

The paper is devoted to the study of asymptotic relations for the function λ γ,σ (β)(z)= β Γ(γ+1−1/β) ∫ 1 ∞ (t β−1) γ−1/βt σ e −zt dt generalising Tricomi confluent hypergeometric function and modified Bessel function of the third kind. The full asymptotic representations for λ γ, σ ( β) ( z) at zero and infinity are established. Applications are given to obtain full asymptotic expansions near zero and infinity for the Liouville fractional integral (I − αf)(x)= 1 Γ(α) ∫ x ∞ f(t) dt (t−x) 1−α (x>0; α∈ C , Re(α)>0) and for the Erdelyi–Kober-type fractional integral (I −;β,η αf)(x)= βx βη Γ(α) ∫ x ∞ t β(1−α−η)−1f(t) dt (t β−x β) 1−α (x>0; α∈ C , ( Re(α)>0) with β>0 and η∈ C of power-exponential function f( t), and for three other fractional integrals.

TRICOMI COHERENT STATES

We construct explicit families of coherent states whose parametrization permits an arbitrarily small deviation from conventional coherent states. They possess the resolution of unity with a positive weight function given as the solution of a Stieltjes power-moment problem, with moments analytically expressible through Tricomi's confluent hypergeometric function. These states are in general squeezed, can be either sub- or super-Poissonian in nature, and induce a deformation of the metric of the complex plane.

Statistical aspects of Tricomi's function and modified Bessel functions of the second kind

Some of the considerable statistical content of modified Bessel functions of the second kind and of Tricomi's confluent hypergeometric function is illustrated. Moment solutions for the parameters of exponential class distribution functions based on both are derived. Unlike the generalised inverse Gaussian, the Tricomi exponential distribution is little known but it emerges that it is of wide applicability, highly flexible and has the gamma distribution as a special case.

Computation of fractional integrals via functions of hypergeometric and Bessel type

The paper is devoted to computation of the fractional integrals of power exponential functions. It is considered a function λ γ, σ ( β) ( z) defined by λ γ,σ (β)(z)= β Γ(γ+1−1/β) ∫ 1 ∞ (t β−1) γ−1/βt σ e −zt dt with positive β and complex γ, σ and z such that Re( γ)>(1/ β)−1 and Re( z)>0. The special cases are discussed when λ γ, σ ( β) ( z) is expressed in terms of the Tricomi confluent hypergeometric function Ψ( a, c; x) and of modified Bessel function of the third kind K γ ( x). Representations of these functions via fractional integrals are proved. The results obtained apply to compute fractional integrals of power exponential functions in terms of λ γ,σ (β)(x), Ψ(a,c;x) and K γ ( x). Examples are considered.

Application of tricomi functions in the eigenvalue analysis of the circular gyrotropic waveguide

The complex Tricomi confluent hypergeometric functions Φ (a, c; x) are introduced as exact functions for propagation of normal TEom modes in circular guiding structures, containing coaxially positioned azimuthally magnetized ferrite layer of double-connected cross-section. An original form of characteristic equation of the ferrite-fiiled coaxial waveguide is derived in terms of the arguments of Tricomi functions. Using the specially tabulated values of the latter, a numerical solution of this equation is accomplished, giving the eigenvalue spectrum and nonreciprocal phase characteristics of the structure. An analysis of its phase shifting capabilities is presented for a variety of geometry and ferrite parameters. The potentialities of the gyrotropic waveguide as digital nonreciprocal phaser, operating in the TE01 mode, are discussed.

Some properties of the circular waveguide with azimuthally magnetized ferrite

A comprehensive analysis of normal rotationally symmetric TE modes in a circular waveguide, filled with ferrite, magnetized azimuthally to remanence by a coaxial switching conductor of finite radius, is presented. The characteristic equation of the structure, derived in terms of Kummer and Tricomi confluent hypergeometric functions of complex parameter and variable, is solved numerically, using specially compiled tables of wave functions. Families of theoretically calculated nonreciprocal phase characteristics of the gyrotropic waveguide are shown in normalized form for the two latched states of remanent magnetization, a variety of ferrite parameters, and different values of switching conductor to waveguide radius ratio. The influence of structure geometry and parameters of anisotropic ferrite on normalized differential phase shift and cutoff frequency spectrum of the TE01 mode is discussed.

Technical memorandum. Kummer and Tricomi functions for analysis of circular gyrotropic waveguides with azimuthal magnetisation

The wave functions for propagation of rotationally symmetric modes in circular gyrotropic waveguides with azimuthal magnetisation (the complex Kummer and Tricomi confluent hypergeometric functions) are studied analytically and numerically. Based on specially compiled tables, families of curves are presented, illustrating the behaviour of both higher transcendental functions. The analysis allows a comprehensive treatment of closed and opened circular guiding structures with radially stratified gyrotropic layers, magnetised azimuthally.

On a Class of Electromagnetic Wave Functions for Propagation Along the Circular Gyrotropic Waveguide

The properties of confluent hypergeometric functions as exact electromagnetic wave fnnctions for propagation in a circular waveguide containing azimuthally magnetized remanent ferrite are investigated. Two different forms of solutions of the propagation problem for angular symmetric transverse electric modes are constructed-one in terms of Kummer and Tricomi confluent hypergeometric functions of complex parameter and variable and a second in terms of Whittaker functions. An evaluation of this class of wave functions is performed to sufficient extent, followed by tabulation of their imaginary zeros, providing computation of eigenvalue spectrum and phase characteristics of the gyrotropic guide.

On the computation of Tricomi's ψ function

Two methods for calculating Tricomi's confluent hypergeometric function are discussed. Both methods are based on recurrence relations. The first method converges like $$\exp ( - \alpha |\lambda |^{1/3} n^{2/3} )for some \alpha > 0$$ and the second like $$\exp ( - \beta |\lambda |^{1/2} n^{1/2} )for some \beta > 0.$$ Several examples are presented.