scholarly journals OKUBO TYPE DIFFERENTIAL EQUATIONS DERIVED FROM HYPERGEOMETRIC FUNCTIONS FD

2021 ◽  
Vol 75 (1) ◽  
pp. 1-21
Author(s):  
Mitsuo KATO
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions

scholarly journals The diagrammatic coaction beyond one loop

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Samuel Abreu ◽  
Ruth Britto ◽  
Claude Duhr ◽  
Einan Gardi ◽  
James Matthew
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Differential Forms ◽  
Feynman Graph ◽  
Loop Order ◽  
Feynman Integrals ◽  
Mathematical Properties ◽  
Multiple Polylogarithms ◽  
Complete Set ◽  
General Tool

Abstract The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

A Class of Generalized Hypergeometric Functions in Several Variables

1992 ◽  
Vol 44 (6) ◽  
pp. 1317-1338 ◽  
Author(s):  
Zhimin Yan
Keyword(s):  
Asymptotic Behavior ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Unique Solution ◽  
Hypergeometric Functions ◽  
Integral Representations ◽  
Gaussian Hypergeometric Function ◽  
Generalized Hypergeometric Functions ◽  
Several Variables

AbstractWe study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

The oblique reflexion of long wireless waves from the ionosphere at the places where the Earth's magnetic field is regarded as vertical

1952 ◽  
Vol 244 (887) ◽  
pp. 469-503 ◽  
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Electromagnetic Waves ◽  
Hypergeometric Functions ◽  
Collision Frequency ◽  
Path Difference ◽  
Stratified Medium ◽  
Graphical Form ◽  
Confluent Hypergeometric Functions ◽  
Ionization Density

The calculation of reflexion coefficients for long wireless waves incident obliquely on the ionosphere requires an exact solution of the differential equations governing the propagation of electromagnetic waves in the ionosphere. Equations are developed for the electromagnetic field in a horizontally stratified medium of varying electron density, the presence of a vertical external magnetic field and also the collision frequency of the electrons with neutral molecules being taken into account. Provided certain inequalities hold amongst these ionospheric characteristics, the ionosphere splits up effectively into two regions, in each of which the differential equations of wave propagation approximate to simpler forms. If a model ionosphere is chosen in which the ionization density increases exponentially with height/and the collision frequency is assumed constant over the range of height responsible for reflexion, the equations for the two regions can be solved exactly. The solution for the lower region is expressed in terms of hypergeometric functions, and that for the upper region in terms of generalized confluent hypergeometric functions. Exact expressions in terms of factorial functions can then be deduced for the reflexion coefficients of both regions separately. Moreover, these coefficients can be combined, with due allowance for the path difference between the two regions, to give the overall reflexion coefficients for the effect of the ionosphere as a whole on an incident wave. A suitable definition is given for the apparent height of reflexion in terms of the phase of the reflected wave. The results of the theory are illustrated in graphical form for a particular model ionosphere approximating to the 'tail’ of a Chapman region, and a brief comparison with experimental observations concludes the paper.

Some Results on the Extended Hypergeometric Function

2020 ◽  
Vol 87 (1-2) ◽  
pp. 70
Author(s):  
Ranjan Kumar Jana ◽  
Bhumika Maheshwari ◽  
Ajay Kumar Shukla
Keyword(s):  
Differential Equations ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Pochhammer Symbol

An attempt is made to define the extended Pochhammer symbol <em>(λ)n,α</em> which leads to an extension of the classical hypergeometric functions. Differential equations and some properties have also been discussed.

scholarly journals MATRIX VALUED SPHERICAL FUNCTIONS ASSOCIATED TO THE THREE DIMENSIONAL HYPERBOLIC SPACE

2002 ◽  
Vol 13 (07) ◽  
pp. 727-784 ◽  
Author(s):  
F. A. GRÜNBAUM ◽  
I. PACHARONI ◽  
J. TIRAO
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hyperbolic Space ◽  
Special Class ◽  
Hypergeometric Functions ◽  
Spherical Function ◽  
Three Dimensional ◽  
Coupled Systems ◽  
Spherical Functions ◽  
Generalized Hypergeometric Functions

The main purpose of this paper is to compute all irreducible spherical functions on [Formula: see text] of arbitrary type [Formula: see text], where K = SU (2). This is accomplished by associating to a spherical function Φ on G a matrix valued function H on the three dimensional hyperbolic space ℍ = G/K. The entries of H are solutions of two coupled systems of ordinary differential equations. By an appropriate twisting involving Hahn polynomials we uncouple one of the systems and express the entries of H in terms of Gauss' functions 2F1. Just as in the compact instance treated in [7], there is a useful role for a special class of generalized hypergeometric functions p+1 Fp.

scholarly journals Dirac–Kähler particle in Riemann spherical space: boson interpretation

2015 ◽  
Vol 93 (11) ◽  
pp. 1427-1433 ◽  
Author(s):  
A.M. Ishkhanyan ◽  
O. Florea ◽  
E.M. Ovsiyuk ◽  
V.M. Red’kov
Keyword(s):  
Angular Momentum ◽  
Energy Spectrum ◽  
General Solution ◽  
Differential Equations ◽  
Hypergeometric Functions ◽  
Total Angular Momentum ◽  
Positive Curvature ◽  
Dirac Particle ◽  
Spherical Space ◽  
Discrete Energy

In the context of the composite boson interpretation, we construct the exact general solution of the Dirac–Kähler equation for the case of the spherical Riemann space of constant positive curvature, for which due to the geometry itself one may expect to have a discrete energy spectrum. In the case of the minimum value of the total angular momentum, j = 0, the radial equations are reduced to second-order ordinary differential equations, which are straightforwardly solved in terms of the hypergeometric functions. For nonzero values of the total angular momentum, however, the radial equations are reduced to a pair of complicated fourth-order differential equations. Employing the factorization approach, we derive the general solution of these equations involving four independent fundamental solutions written in terms of combinations of the hypergeometric functions. The corresponding discrete energy spectrum is then determined via termination of the involved hypergeometric series, resulting in quasi-polynomial wave-functions. The constructed solutions lead to notable observations when comparing with those for the ordinary Dirac particle. The energy spectrum for the Dirac–Kähler particle in spherical space is much more complicated. Its structure substantially differs from the one for the Dirac particle because it consists of two energy level series in parallel, each of which is twice degenerate. Besides, none of the two separate series coincides with the series for the Dirac particle. Thus, the Dirac–Kähler field cannot be interpreted as a system of four Dirac fermions. Additional arguments supporting this conclusion are discussed.

The equality of the moduli of certain ratios occurring in the connexion formulae of solutions of some transcendental differential equations

1970 ◽  
Vol 67 (2) ◽  
pp. 347-361 ◽  
Author(s):  
J. Heading
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Explicit Solutions ◽  
Isotropic Plasma ◽  
General Solutions ◽  
Propagation Of Waves

AbstractThe differential equations governing the propagation of waves of electric and magnetic types in a plane stratified isotropic plasma are suitably generalized, and we investigate the possibility of models for which the moduli of the reflexion coefficients are identical for the two modes. First, the models are examined without the necessity of finding general solutions, and, secondly, by using the circuit relations for the hypergeometric functions occurring in the explicit solutions.

Sign in / Sign up

Export Citation Format

Share Document