26 research outputs found
Differential Recursion Relations for Laguerre Functions on Hermitian Matrices
In our previous papers \cite{doz1,doz2} we studied Laguerre functions and
polynomials on symmetric cones . The Laguerre functions
, , form an orthogonal
basis in and are related via the Laplace
transform to an orthogonal set in the representation space of a highest weight
representations of the automorphism group
corresponding to a tube domain . In this article we consider the
case where is the space of positive definite Hermitian matrices and
. We describe the Lie algebraic realization of
acting in and use that to determine explicit
differential equations and recurrence relations for the Laguerre functions
Differential Recursion Relations for Laguerre Functions on Symmetric Cones
Let be a symmetric cone and the corresponding simple Euclidean
Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of
generalized Laguerre functions on that generalize the classical
Laguerre functions on . This family forms an orthogonal basis for
the subspace of -invariant functions in , where
is a certain measure on the cone and where is the group of
linear transformations on that leave the cone invariant and fix
the identity in . The space supports a highest
weight representation of the group of holomorphic diffeomorphisms that act
on the tube domain In this article we give an explicit
formula for the action of the Lie algebra of and via this action determine
second order differential operators which give differential recursion relations
for the generalized Laguerre functions generalizing the classical creation,
preservation, and annihilation relations for the Laguerre functions on
The generalized Segal-Bargmann transform and special functions
Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner-Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions
Variation in Quality of Diabetes Care at the Levels of Patient, Physician, and Clinic
Introduction: We studied variance in glycated hemoglobin (HbA1c) values among adults with diabetes to identify variation in quality of diabetes care at the levels of patient, physician, and clinic, and to identify which levels contribute the most to variation and which variables at each level are related to quality of diabetes care. Methods: Study subjects were 120 primary care physicians and their 2589 eligible adult patients with diabetes seen at 18 clinics. The dependent variable was HbA1c values recorded in clinical databases. Multivariate hierarchical models were used to partition variation in HbA1c values across the levels of patient, physician, or clinic and to identify significant predictors of HbA1c at each level. Results: More than 95% of variance in HbA1c values was attributable to the patient level. Much less variance was seen at the physician and clinic level. Inclusion of patient and physician covariates did not substantially change this pattern of results. Intensification of pharmacotherapy (t = −7.40, P \u3c .01) and patient age (t = 2.10, P \u3c .05) were related to favorable change in HbA1c. Physician age, physician specialty, number of diabetes patients per physician, patient comorbidity, and clinic assignment did not predict change in HbA1c value. The overall model with covariates explained 11.8% of change in HbA1c value over time. Conclusion: These data suggest that most variance in HbA1c values is attributable to patient factors, although physicians play a major role in some patient factors (e.g., intensification of medication). These findings may lead to more effective care-improvement strategies and accountability measures
Unitary Representations of Lie Groups with Reflection Symmetry
We consider the following class of unitary representations of some
(real) Lie group which has a matched pair of symmetries described as
follows: (i) Suppose has a period-2 automorphism , and that the
Hilbert space carries a unitary operator such that (i.e., selfsimilarity). (ii) An added symmetry is implied
if further contains a closed subspace having
a certain order-covariance property, and satisfying the -restricted positivity: , ,
where is the inner product in . From
(i)--(ii), we get an induced dual representation of an associated dual group
. All three properties, selfsimilarity, order-covariance, and positivity,
are satisfied in a natural context when is semisimple and hermitean; but
when is the -group, or the Heisenberg group, positivity is
incompatible with the other two axioms for the infinite-dimensional irreducible
representations. We describe a class of , containing the latter two, which
admits a classification of the possible spaces satisfying the axioms of selfsimilarity and order-covariance.Comment: 49 pages, LaTeX article style, 11pt size optio
Laguerre polynomials, restriction principle, and holomorphic representations of SL(2, ℝ)
The restriction principle is used to implement a realization of the holomorphic representations of SL(2, ℝ) on L2 (ℝ+, tα dt) by way of the standard upper half plane realization. The resulting unitary equivalence establishes a correspondence between functions that transform according to the character θ ↔ e-i(2n+α+1)θ under rotations and the Laguerre polynomials. The standard recursion relations amongst Laguerre polynomials are derived from the action of the Lie algebra
LAGUERRE FUNCTIONS ON SYMMETRIC CONES AND RECURSION RELATIONS IN THE REAL CASE
Abstract. In this article we derive differential recursion relations for the Laguerre functions on the cone Ω of positive definite real matrices. The highest weight representations of the group Sp(n, R) play a fundamental role. Each such representation acts on a Hilbert space of holomorphic functions on the tube domain Ω + iSym(n, R). We then use the Laplace transform to carry the Lie algebra action over to L 2 (Ω, dµν). The differential recursion relations result by restricting to a distinguished three dimensional subalgebra, which is isomorphic to sl(2, R)