10,309 research outputs found
A note on generalized hypergeometric functions, KZ solutions, and gluon amplitudes
Some aspects of Aomoto's generalized hypergeometric functions on Grassmannian
spaces are reviewed. Particularly, their integral representations
in terms of twisted homology and cohomology are clarified with an example of
the case which corresponds to Gauss' hypergeometric functions. The
cases of in general lead to -point solutions of the
Knizhnik-Zamolodchikov (KZ) equation. We further analyze the
Schechtman-Varchenko integral representations of the KZ solutions in relation
to the cases. We show that holonomy operators of the so-called
KZ connections can be interpreted as hypergeometric-type integrals. This result
leads to an improved description of a recently proposed holonomy formalism for
gluon amplitudes. We also present a (co)homology interpretation of Grassmannian
formulations for scattering amplitudes in super Yang-Mills
theory.Comment: 51 pages; v2. reference added; v3. minor corrections, published
versio
Classical elliptic hypergeometric functions and their applications
General theory of elliptic hypergeometric series and integrals is outlined.
Main attention is paid to the examples obeying properties of the "classical"
special functions. In particular, an elliptic analogue of the Gauss
hypergeometric function and some of its properties are described. Present
review is based on author's habilitation thesis [Spi7] containing a more
detailed account of the subject.Comment: 42 pages, typos removed, references update
Generalized photon-added associated hypergeometric coherent states: characterization and relevant properties
This paper presents the construction of a new set of generalized photon-added
coherent states related to associated hypergeometric functions introduced in
our previous work (Hounkonnou M N and Sodoga K, 2005, J. Phys. A: Math. Gen 38,
7851). These states satisfy all required mathematical and physical properties.
The associated Stieltjes power-moment problem is explicitly solved by using
Meijer's G-function and the Mellin inversion theorem. Relevant quantum optical
and thermal characteristics are investigated. The formalism is applied to
particular cases of the associated Hermite, Laguerre, Jacobi polynomials and
hypergeometric functions. Their corresponding states exhibit sub-Poissonian
photon number statistics
Aspects of elliptic hypergeometric functions
General elliptic hypergeometric functions are defined by elliptic
hypergeometric integrals. They comprise the elliptic beta integral, elliptic
analogues of the Euler-Gauss hypergeometric function and Selberg integral, as
well as elliptic extensions of many other plain hypergeometric and
-hypergeometric constructions. In particular, the Bailey chain technique,
used for proving Rogers-Ramanujan type identities, has been generalized to
integrals. At the elliptic level it yields a solution of the Yang-Baxter
equation as an integral operator with an elliptic hypergeometric kernel. We
give a brief survey of the developments in this field.Comment: 15 pp., 1 fig., accepted in Proc. of the Conference "The Legacy of
Srinivasa Ramanujan" (Delhi, India, December 2012
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