77 research outputs found
The vector-valued big q-Jacobi transform
Big -Jacobi functions are eigenfunctions of a second order -difference
operator . We study as an unbounded self-adjoint operator on an
-space of functions on with a discrete measure. We describe
explicitly the spectral decomposition of using an integral transform
with two different big -Jacobi functions as a kernel, and we
construct the inverse of .Comment: 35 pages, corrected an error and typo
Wilson function transforms related to Racah coefficients
The irreducible -representations of the Lie algebra consist of
discrete series representations, principal unitary series and complementary
series. We calculate Racah coefficients for tensor product representations that
consist of at least two discrete series representations. We use the explicit
expressions for the Clebsch-Gordan coefficients as hypergeometric functions to
find explicit expressions for the Racah coefficients. The Racah coefficients
are Wilson polynomials and Wilson functions. This leads to natural
interpretations of the Wilson function transforms. As an application several
sum and integral identities are obtained involving Wilson polynomials and
Wilson functions. We also compute Racah coefficients for U_q(\su(1,1)), which
turn out to be Askey-Wilson functions and Askey-Wilson polynomials.Comment: 48 page
On a pair of difference equations for the type orthogonal polynomials and related exactly-solvable quantum systems
We introduce a pair of novel difference equations, whose solutions are
expressed in terms of Racah or Wilson polynomials depending on the nature of
the finite-difference step. A number of special cases and limit relations are
also examined, which allow to introduce similar difference equations for the
orthogonal polynomials of the and types. It is shown that
the introduced equations allow to construct new models of exactly-solvable
quantum dynamical systems, such as spin chains with a nearest-neighbour
interaction and fermionic quantum oscillator models.Comment: 8 pages, to be published in Springer Proceedings in Mathematics &
Statistic
A solution to the Al-Salam--Chihara moment problem
We study the -hypergeometric difference operator on a particular
Hilbert space. In this setting can be considered as an extension of the
Jacobi operator for -Al-Salam--Chihara polynomials. Spectral analysis
leads to unitarity and an explicit inverse of a -analog of the Jacobi
function transform. As a consequence a solution of the Al-Salam--Chihara
indeterminate moment problem is obtained.Comment: 22 page
LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials
For little q-Jacobi polynomials and q-Hahn polynomials we give particular
q-hypergeometric series representations in which the termwise q=0 limit can be
taken. When rewritten in matrix form, these series representations can be
viewed as LU factorizations. We develop a general theory of LU factorizations
related to complete systems of orthogonal polynomials with discrete
orthogonality relations which admit a dual system of orthogonal polynomials.
For the q=0 orthogonal limit functions we discuss interpretations on p-adic
spaces. In the little 0-Jacobi case we also discuss product formulas.Comment: changed title, references updated, minor changes matching the version
to appear in Ramanujan J.; 22 p
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
Quantum hypercomputation based on the dynamical algebra su(1,1)
An adaptation of Kieu's hypercomputational quantum algorithm (KHQA) is
presented. The method that was used was to replace the Weyl-Heisenberg algebra
by other dynamical algebra of low dimension that admits infinite-dimensional
irreducible representations with naturally defined generalized coherent states.
We have selected the Lie algebra , due to that this algebra
posses the necessary characteristics for to realize the hypercomputation and
also due to that such algebra has been identified as the dynamical algebra
associated to many relatively simple quantum systems. In addition to an
algebraic adaptation of KHQA over the algebra , we
presented an adaptations of KHQA over some concrete physical referents: the
infinite square well, the infinite cylindrical well, the perturbed infinite
cylindrical well, the P{\"o}sch-Teller potentials, the Holstein-Primakoff
system, and the Laguerre oscillator. We conclude that it is possible to have
many physical systems within condensed matter and quantum optics on which it is
possible to consider an implementation of KHQA.Comment: 25 pages, 1 figure, conclusions rewritten, typing and language errors
corrected and latex format changed minor changes elsewhere and
Empowering Kanban through TPS-principles – an empirical analysis of the Toyota Production System
Montagem e acurácia de um sistema experimental de pesagem para calibração de sensores de umidade do solo
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