110 research outputs found
q-Ultraspherical polynomials for q a root of unity
Properties of the -ultraspherical polynomials for being a primitive
root of unity are derived using a formalism of the algebra. The
orthogonality condition for these polynomials provides a new class of
trigonometric identities representing discrete finite-dimensional analogs of
-beta integrals of Ramanujan.Comment: 7 pages, LATE
Jordan algebras and orthogonal polynomials
We illustrate how Jordan algebras can provide a framework for the
interpretation of certain classes of orthogonal polynomials. The big -1 Jacobi
polynomials are eigenfunctions of a first order operator of Dunkl type. We
consider an algebra that has this operator (up to constants) as one of its
three generators and whose defining relations are given in terms of
anticommutators. It is a special case of the Askey-Wilson algebra AW(3). We
show how the structure and recurrence relations of the big -1 Jacobi
polynomials are obtained from the representations of this algebra. We also
present ladder operators for these polynomials and point out that the big -1
Jacobi polynomials satisfy the Hahn property with respect to a generalized
Dunkl operator.Comment: 11 pages, 30 reference
More on the q-oscillator algebra and q-orthogonal polynomials
Properties of certain -orthogonal polynomials are connected to the
-oscillator algebra. The Wall and -Laguerre polynomials are shown to
arise as matrix elements of -exponentials of the generators in a
representation of this algebra. A realization is presented where the continuous
-Hermite polynomials form a basis of the representation space. Various
identities are interpreted within this model. In particular, the connection
formula between the continuous big -Hermite polynomials and the continuous
-Hermite polynomials is thus obtained, and two generating functions for
these last polynomials are algebraically derived
Hypergeometric Orthogonal Polynomials with respect to Newtonian Bases
We introduce the notion of ''hypergeometric'' polynomials with respect to Newtonian bases. We find the necessary and sufficient conditions for the polynomials Pn(x) to be orthogonal. For the special cases where the sets λn correspond to the classical grids, we find the complete solution to these conditions and observe that it leads to the most general Askey-Wilson polynomials and their special and degenerate classes
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