On the Generalized Confluent Hypergeometric Function and Its Application

2000 Mathematics Subject Classification: 26A33, 33C20

A MATLAB program for the computation of the confluent hypergeometric function Φ2

We here present a sample MATLAB program for the numerical evaluation of the confluent hypergeometric function Φ2. This program is based on the calculation of the inverse Laplace transform using the algorithm suggested by Simon and Alouini in their reference textbook [1].

Asymptotic and structural properties of special cases of the Wright function arising in probability theory

This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function 1Ψ1(ρ, k; ρ, 0; x) = X∞ n=0 Γ(k + ρn) Γ(ρn) x n n! (|x| < ∞) when the parameter ρ ∈ (−1, 0)∪(0, ∞) and the argument x is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter k is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of ρ. The asymptotics of 1Ψ1(ρ, k; ρ, 0; x) are obtained under numerous assumptions on the behavior of the arguments k and x when the parameter ρ is both positive and negative. We also provide some integral representations and structural properties involving the ‘reduced’ Wright function 0Ψ1(−−; ρ, 0; x) with ρ ∈ (−1, 0) ∪ (0, ∞), which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions 0Ψ1(−−;±ρ, 0; ·) and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.

Certain Properties of Fractional Calculus Operators Associated with Generalized Mittag-Leffler Function

Mathematics Subject Classification: 26A33, 33E12, 33C20.

Fractional Extensions of Jacobi Polynomials and Gauss Hypergeometric Function

2000 Mathematics Subject Classification: 26A33, 33C45

T.E. wave in inhomogeneous magnetized plasma-filled cylindrical waveguide

The T. E. wave in cylindrical wavegulde filled with inhomogeneous plasma immersed in the external uniform longitudinal magnetic field is investigated. The analytic solution expressed in polynomial formed by cutting the confluent hypergeometric function is obtained. Furthermore the eigenfrequency of T. E. wave is obtained.

Trapping of a particle in a short-range harmonic potential well

Eigenstates of a particle in a localized and unconfined harmonic potential well are investigated. Effects due to the variation of the potential parameters as well as certain results from asymptotic expansions are discussed. © 2012 Springer Science+Business Media, LLC.

On the Generalized Associated Legendre Functions

Mathematics Subject Classification: 33C60, 33C20, 44A15

On Some Generalizations of Classical Integral Transforms

MSC 2010: 44A15, 44A20, 33C60

Krätzel Function as a Function of Hypergeometric Type

2000 Mathematics Subject Classification: 33C60, 33C20, 44A15

Completeness of the Coulomb wave functions in quantum mechanics

We give an explicit, direct, and fairly elementary proof that the radial energy eigenfunctions for the hydrogen atom in quantum mechanics, bound and scattering states included, form a complete set. The proof uses only some properties of the confluent hypergeometric functions and the Cauchy residue theorem from analytic function theory; therefore it would form useful supplementary reading for a graduate course on quantum mechanics.

SZEGO POLYNOMIALS FROM HYPERGEOMETRIC FUNCTIONS

Szego polynomials with respect to the weight function w(theta) = e(eta theta)[sin(theta/2)](2 lambda), where eta, lambda is an element of R and lambda > -1/2 are considered. Many of the basic relations associated with these polynomials are given explicitly. Two sequences of para-orthogonal polynomials with explicit relations are also given.

Further Generalization of Kobayashi's Gamma Function

In this paper, we introduce a further generalization of the gamma function involving Gauss hypergeometric function 2F1 (a, b; c; z)

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Effective dielectric response of composites witli graded material

The effective dielectric response of linear composites containing graded material is investigated under an applied electric field Eo. For the cylindrical inclusion with gradient dielectric function, epsilon(i)(r) = b + cr, randomly embedded in a host with dielectric constant epsilon(m), we have obtained the exact solution of local electric potential of the composite media regions, which obeys a linear constitutive relation D = epsilonE, using hypergeometric function. In dilute limit, we have derived the effective dielectric response of the linear composite media. Furthermore, for larger volume fraction, the formulas of effective dielectric response of the graded composite media, are given.