Convolutions for orthogonal polynomials from Lie and quantum algebra representations

The interpretation of the Meixner-Pollaczek, Meixner and Laguerre polynomials as overlap coefficients in the positive discrete series representations of the Lie algebra su(1,1) and the Clebsch-Gordan decomposition leads to generalisations of the convolution identities for these polynomials. Using the Racah coefficients convolution identities for continuous Hahn, Hahn and Jacobi polynomials are obtained. From the quantised universal enveloping algebra for su(1,1) convolution identities for the Al-Salam and Chihara polynomials and the Askey-Wilson polynomials are derived by using the Clebsch-Gordan and Racah coefficients. For the quantised universal enveloping algebra for su(2) q-Racah polynomials are interpreted as Clebsch-Gordan coefficients, and the linearisation coefficients for a two-parameter family of Askey-Wilson polynomials are derived.Comment: AMS-TeX, 31 page

Fock representations of the Lie superalgebra q(n+1)

For the Lie superalgebra $q(n+1)$ a description is given in terms of creation and annihilation operators, in such a way that the defining relations of $q(n+1)$ are determined by quadratic and triple supercommutation relations of these operators. Fock space representations $V_p$ of $q(n+1)$ are defined by means of these creation and annihilation operators. These new representations are introduced as quotient modules of some induced module of $q(n+1)$. The representations $V_p$ are not graded, but they possess a number of properties that are of importance for physical applications. For $p$ a positive integer, these representations $V_p$ are finite-dimensional, with a unique highest weight (of multiplicity 1). The Hermitian form that is consistent with the natural adjoint operation on $q(n+1)$ is shown to be positive definite on $V_p$. For $q(2)$ these representations are ``dispin''. For the general case of $q(n+1)$, many structural properties of $V_p$ are derived.Comment: 24 pages, LaTeX file, small corrections done; to appear in J. Phys. A: Math. Ge

What research we no longer need in neurodegenerative disease at the end of life : The case of research in dementia

A complete silence. That was what we got back from the European experts who had been energetically discussing research priorities in palliative care in neurodegenerative disease (ND) until a short while ago.1 The chair, an entertaining professor with good manners, must have felt the unease and quickly refocused the group to their task. But, wasn’t this the best question of all day? What research we no longer need? As scientists able to consider different perspectives, shouldn’t we have some idea of what research is, by contrast, no longer necessary? Palliative care research and research with people who have ND and are at the end of their life is, by definition, difficult. Making choices is a sensitive issue, but funds are limited. Therefore, we take a counterpoint to the research agenda recently reported by European Union (EU) Joint Programme – Neurodegenerative Disease Research (JPND),1 and consider whether there are studies we no longer need or are low priority, taking the example of dementiaPeer reviewedFinal Accepted Versio

Influence of strongly coupled, hidden scalars on Higgs signals

To investigate the possible effects of a light hidden sector on Higgs boson detection, we discuss a model of scalar singlets coupled to the Standard Model. The model effectively makes the Higgs width a free parameter due to additional invisible decay modes. This width can become arbitrarily large. Theoretical and experimental bounds on model parameters are presented. It is shown, how Standard Model predictions change and that in the case of large coupling, Higgs signals will be diluted. We study, to which extent such a strongly coupled, hidden sector can be excluded by present and future Higgs search experiments.Comment: 22 pages, Latex, 12 figures embedded with epsf.sty and epsfig.sty, to appear in Z. Phys.