Tricomi Functions Research Articles

Generalized Tricomi and Hermite–Tricomi functions

General classes of Tricomi and Hermite–Tricomi functions are introduced by exploiting properties of an iterated isomorphism, related to the so-called Laguerre-type exponentials, and we mainly consider the properties of the general classes of 3-variable 2-index Tricomi functions and 2-index 4-variable 1-parameter Hermite–Tricomi functions.

GENERATING RELATIONS INVOLVING 3-VARIABLE 2-PARAMETER TRICOMI FUNCTIONS USING LIE-ALGEBRAIC TECHNIQUES

This paper is an attempt to stress the usefulness of the multi- variable special functions. In this paper, we derive generating relations involving 3-variable 2-parameter Tricomi functions by using Lie-algebraic techniques. Further we derive certain new and known generating relations involving other forms of Tricomi and Bessel functions as applications.

Lie-theoretic generating relations involving multi-variable Hermite–Tricomi functions

This paper is an attempt to stress the usefulness of multi-variable special functions. We derive generating relations involving multi-variable Hermite–Tricomi functions by using the Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Hermite–Tricomi functions as applications.

Generating relations of Tricomi and Hermite–Tricomi functions using Lie algebra representation

This paper is an attempt to stress the usefulness of the multi-variable special functions. In this paper, we derive generating relations involving multi-variable Tricomi and Hermite–Tricomi functions by using Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Tricomi, Hermite–Tricomi and Bessel functions as applications.

Generating Relations of Hermite–Tricomi Functions Using a Representation of Lie Algebra 𝑇3

Abstract Motivated by recent studies of the properties of new classes of polynomials constructed in terms of quasi-monomials, certain generating relations involving Hermite–Tricomi functions are obtained. To accomplish this we use the representation 𝑄(𝑤, 𝑚0) of the 3-dimensional Lie algebra 𝑇3. Some special cases are also discussed.

Monotonicity Properties of Determinants of Special Functions

We prove the absolute monotonicity or complete monotonicity of some determinant functions whose entries involve $$\psi^{(m)}(x)=({d^m}/{dx^m}) [\Gamma'(x)/\Gamma(x)],$$ modified Bessel functions Iν, Kν, the confluent hypergeometric function Φ, and the Tricomi function Ψ. Our results recover and generalize some known determinantal inequalities. We also show that a certain determinant formed by the Fibonacci numbers are nonnegative while determinants involving Hermite polynomials of imaginary arguments are shown to be completely monotonic functions.

ON CERTAIN PROPERTIES OF SOME GENERALIZED SPECIAL FUNCTIONS

In this paper, we derive a result concerning eigenvector for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. The results given by Radulescu, Mandal and authors follow as special cases of this result. Further using these results, we deduce certain properties of generalized Hermite polynomials and Hermite Tricomi functions.

Tricomi and Kummer functions in occurrence, waiting time and exceedance statistics

Three and four parameter distributions of events and intervals based on the Kummer and Tricomi functions are shown to be statistically tractable. The Kummer distributions cover events less as well as more clustered than the Poisson while the Tricomi distributions have two and three parameter negative binomials as special cases. This flexibility and the moment relations between the parameters make them attractive in cases where the Poisson distribution proves inadequate.

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

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

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.

Operational Identities and Properties of Ordinary and Generalized Special Functions

The theory of Hermite, Laguerre, and of the associated generating functions is reformulated within the framework of an operational formalism. This point of view provides more efficient tools which allow the straightforward derivation of a wealth of new and old identities. In this paper a central role is played by negative derivative operators and by their link with the Tricomi functions and the generalized Laguerre polynomials.

Modeling service–time distributions with non–exponential tails:beta mixtures of exponentials

Motivated by interest in probability density functions (pdf's) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdf's. These classes include distributions introduced by Boxma and Cohen (1997) and Gaver and Jacobs (1998) to study queues with long-tail service-time distributions. When the standard beta pdf is used as the mixing pdf, we obtain pdf's with an exponentially damped power tail, i.e. as . This pdf decays exponentially, but analysis is complicated by the power term. When the beta pdf of the second kind is used as the mixing pdf, we obtain pdf's with a power tail, i.e. as . We obtain explicit representations for the cumulative distributions functions, Laplace transforms, moments and asymptotics by exploiting connections to the Tricomi function. Properties of the power-tail class can be deduced directly from properties of the other class, because the power-tail pdf's are undamped versions of the other pdf's. The power-tail class can also be represented as gamma mixtures of Pareto pdf's. Both classes of pdf's have simple explicit Laguerre-series expansions

The Tricomi Time Correlation Function

AbstractA time correlation function based on the Tricomi function with two shape parameters is proposed For special values of these parameters the Debye and Cole‐Davidson functions are recovered. The Tricomi function can be Fourier transformed analytically and an equation for the spectral density is derived.

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.

TE 01 -mode propagation in a circular guide containing azimuthally magnetised remanent ferrite

A new analytical technique based on Tricomi function formalism is developed and applied to study and interpret the properties of a circular guide, filled with azimuthally magnetised remanent ferrite. The nonreciprocal phase shifting and magnetically controlled cutoff experienced by the TE01 mode in the guide can be used in designing microwave ferrite control components.

Computation of parabolic cylinder functions by means of a tricomi expansion

Fast convergence expansion of the parabolic cylinder functions U( a, x), V( a, x), W( a, ± x) is obtained in terms of the Tricomi functions E v ( z). The numerical results are quite accurate for a large interval of values of “ a” and for | x| ⪅ 7. Tables are given for U and V in order to compare our results with other recent works on the same functions.

Special Functions, Stieltjes Transforms and Infinite Divisibility

We establish the complete monotonicity of several quotients of Whittaker (Tricomi) functions and of parabolic cylinder functions. These results are used to show that the F distribution of any positive degrees of freedom (including fractional) is infinitely divisible and self-decomposable. We also prove the infinite divisibility of several related distributions, including the square of a gamma variable. We also prove that $x^{{{(\nu - \mu )} /2}} {{I_\mu (\sqrt x )} /{I_\nu (\sqrt x )}}$ is a completely monotonic function of x when $\mu > \nu > - 1$. This result and the complete monotonicity of $x^{{{(\nu - \mu )} /2}} {{K_\nu (\sqrt x )} /{K_\mu (\sqrt x )}}$, $\mu > \nu > - 1$, are used to introduce two new continuous infinitely divisible probability distributions. The limiting cases contain the reciprocal of a gamma distribution and a distribution whose probability density function is a “generalized” theta function. The first distribution is used as a mixing distribution to introduce a new, two parameter, symmetric, infinitely divisible probability distribution on the real line, which contains the Student t distribution as a limiting case. We also establish the complete monotonicity of ${{K_\nu (b\sqrt x )} /{K_\nu (a\sqrt x )}}$ and ${{I_\nu (a\sqrt x )} /{I_\nu (b\sqrt x )}}$ for $b > a > 0$ and $\nu > - 1$. We also obtain some results on the zeros of combinations of modified Bessel functions.