173,654 research outputs found
The homogeneous q-difference operator and the related polynomials
We create the homogeneous q-difference operator Ee(a, b; θ) as an extension of the exponential operator E(bθ). A new polynomials hn(a, b, x|q−1) are defined as an extension of the q−1-Rogers-Szegö polynomial hn(a, b|q−1). We provide an operator proof of the generating function and its extension, Rogers formula and the invers linearization formula, and Mehler’s formula for the polynomials hn(a, b|q−1). The generating function and its extension, Rogers formula and the invers linearization formula, and Mehler’s formula for the polynomials hn(a, b|q−1) are deduced by giving special values to parameters of a new polynomial hn(a, b, x|q−1).Publisher's Versio
Semi-classical Orthogonal Polynomial Systems on Non-uniform Lattices, Deformations of the Askey Table and Analogs of Isomonodromy
A -semi-classical weight is one which satisfies a particular
linear, first order homogeneous equation in a divided-difference operator
. It is known that the system of polynomials, orthogonal with
respect to this weight, and the associated functions satisfy a linear, first
order homogeneous matrix equation in the divided-difference operator termed the
spectral equation. Attached to the spectral equation is a structure which
constitutes a number of relations such as those arising from compatibility with
the three-term recurrence relation. Here this structure is elucidated in the
general case of quadratic lattices. The simplest examples of the
-semi-classical orthogonal polynomial systems are precisely those
in the Askey table of hypergeometric and basic hypergeometric orthogonal
polynomials. However within the -semi-classical class it is
entirely natural to define a generalisation of the Askey table weights which
involve a deformation with respect to new deformation variables. We completely
construct the analogous structures arising from such deformations and their
relations with the other elements of the theory. As an example we treat the
first non-trivial deformation of the Askey-Wilson orthogonal polynomial system
defined by the -quadratic divided-difference operator, the Askey-Wilson
operator, and derive the coupled first order divided-difference equations
characterising its evolution in the deformation variable. We show that this
system is a member of a sequence of classical solutions to the
-Painlev\'e system.Comment: Submitted to Duke Mathematical Journal on 5th April 201
Some New Generating Functions for q
We obtain some new generating functions for q-Hahn polynomials and give their proofs based on the homogeneous q-difference operator
Projections with fixed difference: a Hopf-Rinow theorem
The set D_A0 , of pairs of orthogonal projections (P,Q) in generic position with fixed difference P−Q=A_0, is shown to be a homogeneous smooth manifold: it is the quotient of the unitary group of the commutant {A_0}′ divided by the unitary subgroup of the commutant {P0,Q0}′, where (P0,Q0) is any fixed pair in D_A0. Endowed with a natural reductive structure (a linear connection) and the quotient Finsler metric of the operator norm, it behaves as a classic Riemannian space: any two pairs in D_A0 are joined by a geodesic of minimal length. Given a base pair (P0,Q0), pairs in an open dense subset of DA0 can be joined to (P0,Q0) by a unique minimal geodesic.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Corach, Gustavo. Universidad de Buenos Aires. Facultad de IngenierÃa; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Recht, Lázaro. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Simon Bolivar.; Venezuel
Raising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice
We complete the construction of raising and lowering operators, given in a
previous work, for the orthogonal polynomials of hypergeometric type on
non-homogeneous lattice, and extend these operators to the generalized
orthogonal polynomials, namely, those difference of orthogonal polynomials that
satisfy a similar difference equation of hypergeometric type.Comment: LaTeX, 19 pages, (late submission to arXiv.org
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