scholarly journals Generalized hypergeometric series for Racah matrices in rectangular representations

2018 ◽  
Vol 33 (04) ◽  
pp. 1850020 ◽  
Author(s):  
A. Morozov
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Mathematical Physics ◽  
Natural Question ◽  
Generalized Hypergeometric Series ◽  
Wilson Polynomials ◽  
The One ◽  
Quantum Dimensions ◽  
Skew Characters ◽  
Trivial Fact

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group [Formula: see text] through the Askey–Wilson polynomials, associated with the [Formula: see text]-hypergeometric functions [Formula: see text]. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary [Formula: see text] — at least for exclusive Racah matrices [Formula: see text]. The natural question then is what substitutes the conventional [Formula: see text]-hypergeometric polynomials when representations are more general? New advances in the theory of matrices [Formula: see text], provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of [Formula: see text]. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

scholarly journals An extension of Pochhammer’s symbol and its application to hypergeometric functions

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 207-215 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Gamma Function ◽  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Special Property ◽  
Convergence Problem ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Series ◽  
General Properties

By using a special property of the gamma function, we first define a productive form of gamma and beta functions and study some of their general properties in order to define a new extension of the Pochhammer symbol. We then apply this extended symbol for generalized hypergeometric series and study the convergence problem with some illustrative examples in this sense. Finally, we introduce two new extensions of Gauss and confluent hypergeometric series and obtain some of their general properties.

Zeros of generalized hypergeometric functions

1973 ◽  
Vol 74 (2) ◽  
pp. 269-276
Author(s):  
A. D'Adda ◽  
R. D'Auria
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
Image Position

In this paper we derive the conditions which have to be satisfied in order to obtain some classes of zeros of the generalized hypergeometric series of the typeThese conditions read:

scholarly journals An extension of Pochhammer’s symbol and its application to hypergeometric functions, II

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6505-6517
Author(s):  
Mohammad Masjed-Jamei ◽  
Gradimir Milovanovic
Keyword(s):  
Hypergeometric Functions ◽  
Hypergeometric Series ◽  
Arbitrary Order ◽  
Beta Function ◽  
Pochhammer Symbol ◽  
Generalized Hypergeometric Functions ◽  
Generalized Hypergeometric Series ◽  
General Properties ◽  
Different Types ◽  
Additive Form

Recently we have introduced a productive form of gamma and beta functions and applied them for generalized hypergeometric series [Filomat, 31 (2017), 207-215]. In this paper, we define an additive form of gamma and beta functions and study some of their general properties in order to obtain a new extension of the Pochhammer symbol. We then apply the new symbol for introducing two different types of generalized hypergeometric functions. In other words, based on the defined additive beta function, we first introduce an extension of Gauss and confluent hypergeometric series and then, based on two additive types of the Pochhammer symbol, we introduce two extensions of generalized hypergeometric functions of any arbitrary order. The convergence of each series is studied separately and some illustrative examples are given in the sequel.

scholarly journals Dissipative Processes and Their Role in the Evolution of Radio Galaxies

Galaxies ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 70 ◽  
Author(s):  
Manel Perucho
Keyword(s):  
Particle Acceleration ◽  
Kinetic Energy ◽  
Radio Galaxies ◽  
Active Galaxies ◽  
Energy Budgets ◽  
Natural Question ◽  
Relativistic Jets ◽  
The One ◽  
Very High

Particle acceleration in relativistic jets, to very high levels of energy, occurs at the expense of the dissipation of magnetic or kinetic energy. Therefore, understanding the processes that can trigger this dissipation is key to the characterization of the energy budgets and particle acceleration mechanisms in action in active galaxies. Instabilities and entrainment are two obvious candidates to trigger dissipation. On the one hand, supersonic, relativistic flows threaded by helical fields, as expected from the standard formation models of jets in supermassive black-holes, are unstable to a series of magnetohydrodynamical instabilities, such as the Kelvin–Helmholtz, current-driven, or possibly the pressure-driven instabilities. Furthermore, in the case of expanding jets, the Rayleigh–Taylor and centrifugal instabilities may also develop. With all these destabilizing processes in action, a natural question is to ask how can some jets keep their collimated structure along hundreds of kiloparsecs. On the other hand, the interaction of the jet with stars and clouds of gas that cross the flow in their orbits around the galactic centers provides another scenario in which kinetic energy can be efficiently converted into internal energy and particles can be accelerated to non-thermal energies. In this contribution, I review the conditions under which these processes occur and their role both in jet evolution and propagation and energy dissipation.

scholarly journals Euler Type Integrals and Integrals in Terms of Extended Beta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Subuhi Khan ◽  
Mustafa Walid Al-Saad
Keyword(s):  
Hypergeometric Series ◽  
Beta Function ◽  
Generalized Hypergeometric Series ◽  
Special Polynomials ◽  
Extended Beta Function

We derive the evaluations of certain integrals of Euler type involving generalized hypergeometric series. Further, we establish a theorem on extended beta function, which provides evaluation of certain integrals in terms of extended beta function and certain special polynomials. The possibility of extending some of the derived results to multivariable case is also investigated.

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