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 $\Omega=H/L$. The Laguerre functions $\ell^{\nu}_{\mathbf{n}}$, $\mathbf{n}\in\mathbf{\Lambda}$, form an orthogonal basis in $L^{2}(\Omega,d\mu_{\nu})^{L}$ and are related via the Laplace transform to an orthogonal set in the representation space of a highest weight representations $(\pi_{\nu}, \mathcal{H}_{\nu})$ of the automorphism group $G$ corresponding to a tube domain $T(\Omega)$. In this article we consider the case where $\Omega$ is the space of positive definite Hermitian matrices and $G=\mathrm{SU}(n,n)$. We describe the Lie algebraic realization of $\pi_{\nu}$ acting in $L^{2}(\Omega,d\mu_{\nu})$ 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 $\Omega$ be a symmetric cone and $V$ the corresponding simple Euclidean Jordan algebra. In \cite{ado,do,do04,doz2} we considered the family of generalized Laguerre functions on $\Omega$ that generalize the classical Laguerre functions on $\mathbb{R}^+$. This family forms an orthogonal basis for the subspace of $L$-invariant functions in $L^2(\Omega,d\mu_\nu)$, where $d\mu_\nu$ is a certain measure on the cone and where $L$ is the group of linear transformations on $V$ that leave the cone $\Omega$ invariant and fix the identity in $\Omega$. The space $L^2(\Omega,d\mu_\nu)$ supports a highest weight representation of the group $G$ of holomorphic diffeomorphisms that act on the tube domain $T(\Omega)=\Omega + iV.$ In this article we give an explicit formula for the action of the Lie algebra of $G$ 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 $\mathbb{R}^+$

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 $\pi$ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau$, and that the Hilbert space $\mathbf{H} (\pi)$ carries a unitary operator $J$ such that $J\pi =(\pi \circ \tau)J$ (i.e., selfsimilarity). (ii) An added symmetry is implied if $\mathbf{H} (\pi)$ further contains a closed subspace $\mathbf{K}_0$ having a certain order-covariance property, and satisfying the $\mathbf{K}_0$-restricted positivity: $\ge 0$, $\forall v\in \mathbf{K}_0$, where $$ is the inner product in $\mathbf{H} (\pi)$. From (i)--(ii), we get an induced dual representation of an associated dual group $G^c$. All three properties, selfsimilarity, order-covariance, and positivity, are satisfied in a natural context when $G$ is semisimple and hermitean; but when $G$ is the $(ax+b)$-group, or the Heisenberg group, positivity is incompatible with the other two axioms for the infinite-dimensional irreducible representations. We describe a class of $G$, containing the latter two, which admits a classification of the possible spaces $\mathbf{K}_0 \subset \mathbf{H} (\pi)$ 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)