23,807 research outputs found
Geometric convexity of the generalized sine and the generalized hyperbolic sine
In the paper, the authors prove that the generalized sine function
and the generalized hyperbolic sine function
are geometrically concave and geometrically convex, respectively. Consequently,
the authors verify a conjecture posed in the paper "B. A. Bhayo and M.
Vuorinen, On generalized trigonometric functions with two parameters, J.
Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at
\url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".Comment: 5 page
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV
We solve the longstanding problem to define a functional characterization of
the spectrum of the transfer matrix associated to the most general spin-1/2
representations of the 6-vertex reflection algebra for general inhomogeneous
chains. The corresponding homogeneous limit reproduces the spectrum of the
Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most
general integrable boundaries. The spectrum is characterized by a second order
finite difference functional equation of Baxter type with an inhomogeneous term
which vanishes only for some special but yet interesting non-diagonal boundary
conditions. This functional equation is shown to be equivalent to the known
separation of variable (SOV) representation hence proving that it defines a
complete characterization of the transfer matrix spectrum. The polynomial
character of the Q-function allows us then to show that a finite system of
equations of generalized Bethe type can be similarly used to describe the
complete transfer matrix spectrum.Comment: 28 page
Logarithmic mean inequality for generalized trigonometric and hyperbolic functions
In this paper we study the convexity and concavity properties of generalized
trigonometric and hyperbolic functions in case of Logarithmic mean.Comment:
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