1,144 research outputs found
On inversion and connection coefficients for basic hypergeometric polynomials
In this paper, we propose a general method to express explicitly the
inversion and the connection coefficients between two basic hypergeometric
polynomial sets. As application, we consider some -orthogonal basic
hypergeometric polynomials and we derive expansion formulae corresponding to
all the families within the -Askey scheme.Comment: 15 page
Orthogonal Polynomials from Hermitian Matrices II
This is the second part of the project `unified theory of classical
orthogonal polynomials of a discrete variable derived from the eigenvalue
problems of hermitian matrices.' In a previous paper, orthogonal polynomials
having Jackson integral measures were not included, since such measures cannot
be obtained from single infinite dimensional hermitian matrices. Here we show
that Jackson integral measures for the polynomials of the big -Jacobi family
are the consequence of the recovery of self-adjointness of the unbounded Jacobi
matrices governing the difference equations of these polynomials. The recovery
of self-adjointness is achieved in an extended Hilbert space on which
a direct sum of two unbounded Jacobi matrices acts as a Hamiltonian or a
difference Schr\"odinger operator for an infinite dimensional eigenvalue
problem. The polynomial appearing in the upper/lower end of Jackson integral
constitutes the eigenvector of each of the two unbounded Jacobi matrix of the
direct sum. We also point out that the orthogonal vectors involving the
-Meixner (-Charlier) polynomials do not form a complete basis of the
Hilbert space, based on the fact that the dual -Meixner polynomials
introduced in a previous paper fail to satisfy the orthogonality relation. The
complete set of eigenvectors involving the -Meixner polynomials is obtained
by constructing the duals of the dual -Meixner polynomials which require the
two component Hamiltonian formulation. An alternative solution method based on
the closure relation, the Heisenberg operator solution, is applied to the
polynomials of the big -Jacobi family and their duals and -Meixner
(-Charlier) polynomials.Comment: 65 pages. Comments, references and table of contents are added. To
appear in J.Math.Phy
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
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