162 research outputs found
Spectral Analysis of Certain Schr\"odinger Operators
The -matrix method is extended to difference and -difference operators
and is applied to several explicit differential, difference, -difference and
second order Askey-Wilson type operators. The spectrum and the spectral
measures are discussed in each case and the corresponding eigenfunction
expansion is written down explicitly in most cases. In some cases we encounter
new orthogonal polynomials with explicit three term recurrence relations where
nothing is known about their explicit representations or orthogonality
measures. Each model we analyze is a discrete quantum mechanical model in the
sense of Odake and Sasaki [J. Phys. A: Math. Theor. 44 (2011), 353001, 47
pages]
Generalized Burchnall-Type Identities for Orthogonal Polynomials and Expansions
Burchnall's method to invert the Feldheim-Watson linearization formula for
the Hermite polynomials is extended to all polynomial families in the
Askey-scheme and its -analogue. The resulting expansion formulas are made
explicit for several families corresponding to measures with infinite support,
including the Wilson and Askey-Wilson polynomials. An integrated version gives
the possibility to give alternate expression for orthogonal polynomials with
respect to a modified weight. This gives expansions for polynomials, such as
Hermite, Laguerre, Meixner, Charlier, Meixner-Pollaczek and big -Jacobi
polynomials and big -Laguerre polynomials. We show that one can find
expansions for the orthogonal polynomials corresponding to the
Toda-modification of the weight for the classical polynomials that correspond
to known explicit solutions for the Toda lattice, i.e., for Hermite, Laguerre,
Charlier, Meixner, Meixner-Pollaczek and Krawtchouk polynomials
Meixner polynomials of the second kind and quantum algebras representing su(1,1)
We show how Viennot's combinatorial theory of orthogonal polynomials may be
used to generalize some recent results of Sukumar and Hodges on the matrix
entries in powers of certain operators in a representation of su(1,1). Our
results link these calculations to finding the moments and inverse polynomial
coefficients of certain Laguerre polynomials and Meixner polynomials of the
second kind. As an immediate consequence of results by Koelink, Groenevelt and
Van Der Jeugt, for the related operators, substitutions into essentially the
same Laguerre polynomials and Meixner polynomials of the second kind may be
used to express their eigenvectors. Our combinatorial approach explains and
generalizes this "coincidence".Comment: several correction
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
Spectral decomposition and matrix-valued orthogonal polynomials
The relation between the spectral decomposition of a self-adjoint operator
which is realizable as a higher order recurrence operator and matrix-valued
orthogonal polynomials is investigated. A general construction of such
operators from scalar-valued orthogonal polynomials is presented. Two examples
of matrix-valued orthogonal polynomials with explicit orthogonality relations
and three-term recurrence relation are presented, which both can be considered
as -matrix-valued analogues of subfamilies of Askey-Wilson
polynomials.Comment: 15 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
Big q-Laguerre and q-Meixner polynomials and representations of the algebra U_q(su(1,1))
Diagonalization of a certain operator in irreducible representations of the
positive discrete series of the quantum algebra U_q(su(1,1)) is studied.
Spectrum and eigenfunctions of this operator are found in an explicit form.
These eigenfunctions, when normalized, constitute an orthonormal basis in the
representation space. The initial U_q(su(1,1))-basis and the basis of
eigenfunctions are interrelated by a matrix with entries, expressed in terms of
big q-Laguerre polynomials. The unitarity of this connection matrix leads to an
orthogonal system of functions, which are dual with respect to big q-Laguerre
polynomials. This system of functions consists of two separate sets of
functions, which can be expressed in terms of q-Meixner polynomials
M_n(x;b,c;q) either with positive or negative values of the parameter b. The
orthogonality property of these two sets of functions follows directly from the
unitarity of the connection matrix. As a consequence, one obtains an
orthogonality relation for q-Meixner polynomials M_n(x;b,c;q) with b<0. A
biorthogonal system of functions (with respect to the scalar product in the
representation space) is also derived.Comment: 15 pages, LaTe
The oscillator model for the Lie superalgebra sh(2|2) and Charlier polynomials
We investigate an algebraic model for the quantum oscillator based upon the
Lie superalgebra sh(2|2), known as the Heisenberg-Weyl superalgebra or "the
algebra of supersymmetric quantum mechanics", and its Fock representation. The
model offers some freedom in the choice of a position and a momentum operator,
leading to a free model parameter gamma. Using the technique of Jacobi
matrices, we determine the spectrum of the position operator, and show that its
wavefunctions are related to Charlier polynomials C_n with parameter gamma^2.
Some properties of these wavefunctions are discussed, as well as some other
properties of the current oscillator model.Comment: Minor changes and some additional references added in version
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
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
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