Variations of Stieltjes-Wigert and q-Laguerre polynomials and their recurrence coefficients

We look at some extensions of the Stieltjes-Wigert weight functions. First we replace the variable x by x^2 in a family of weight functions given by Askey in 1989 and we show that the recurrence coefficients of the corresponding orthogonal polynomials can be expressed in terms of a solution of the q-discrete Painlev\'e III equation. Next we consider the q-Laguerre or generalized Stieltjes-Wigert weight functions with a quadratic transformation and derive recursive equations for the recurrence coefficients of the orthogonal polynomials. These turn out to be related to the q-discrete Painlev\'e V equation. Finally we also consider the little q-Laguerre weight with a quadratic transformation and show that the recurrence coefficients of the orthogonal polynomials are again related to q-discrete Painlev\'e V.Comment: 19 page

Bioinformatics analysis reveals biophysical and evolutionary insights into the 3-nitrotyrosine post-translational modification in the human proteome

Protein 3-nitrotyrosine is a post-translational modification that commonly arises from the nitration of tyrosine residues. This modification has been detected under a wide range of pathological conditions and has been shown to alter protein function. Whether 3-nitrotyrosine is important in normal cellular processes or is likely to affect specific biological pathways remains unclear. Using GPS-YNO2, a recently described 3-nitrotyrosine prediction algorithm, a set of predictions for nitrated residues in the human proteome was generated. In total, 9.27 per cent of the proteome was predicted to be nitratable (27 922/301 091). By matching the predictions against a set of curated and experimentally validated 3-nitrotyrosine sites in human proteins, it was found that GPS-YNO2 is able to predict 73.1 per cent (404/553) of these sites. Furthermore, of these sites, 42 have been shown to be nitrated endogenously, with 85.7 per cent (36/42) of these predicted to be nitrated. This demonstrates the feasibility of using the predicted dataset for a whole proteome analysis. A comprehensive bioinformatics analysis was subsequently performed on predicted and all experimentally validated nitrated tyrosine. This found mild but specific biophysical constraints that affect the susceptibility of tyrosine to nitration, and these may play a role in increasing the likelihood of 3-nitrotyrosine to affect processes, including phosphorylation and DNA binding. Furthermore, examining the evolutionary conservation of predicted 3-nitrotyrosine showed that, relative to non-nitrated tyrosine residues, 3-nitrotyrosine residues are generally less conserved. This suggests that, at least in the majority of cases, 3-nitrotyrosine is likely to have a deleterious effect on protein function and less likely to be important in normal cellular function. © 2013 The Authors.Link_to_subscribed_fulltex

The fate and lifespan of human monocyte subsets in steady state and systemic inflammation.

In humans, the monocyte pool comprises three subsets (classical, intermediate, and nonclassical) that circulate in dynamic equilibrium. The kinetics underlying their generation, differentiation, and disappearance are critical to understanding both steady-state homeostasis and inflammatory responses. Here, using human in vivo deuterium labeling, we demonstrate that classical monocytes emerge first from marrow, after a postmitotic interval of 1.6 d, and circulate for a day. Subsequent labeling of intermediate and nonclassical monocytes is consistent with a model of sequential transition. Intermediate and nonclassical monocytes have longer circulating lifespans (∼4 and ∼7 d, respectively). In a human experimental endotoxemia model, a transient but profound monocytopenia was observed; restoration of circulating monocytes was achieved by the early release of classical monocytes from bone marrow. The sequence of repopulation recapitulated the order of maturation in healthy homeostasis. This developmental relationship between monocyte subsets was verified by fate mapping grafted human classical monocytes into humanized mice, which were able to differentiate sequentially into intermediate and nonclassical cells

Discrete Painlevé Equations and Orthogonal Polynomials.

We bestuderen het verband tussen bepaalde semi-klassieke orthogonale veeltermen en discrete Painlev\'evergelijkingen. De hoofdrol in dit verband is weggelegd voor de recursieco\"effici\"enten van de drietermsrecursierelatie waaraan alle rijen van orthogonale veeltermen voldoen. Concreet hebben we aangetoond dat de recursieco\"effici\"enten van semi-klassieke Laguerre, Stieltjes-Wigert, $q$-Laguerre en kleine $q$-Laguerre veeltermen, na een simpele transformatie, voldoen aan discrete Painlev\'evergelijkingen. Voor verschillende andere rijen van semiklassieke veeltermen hebben we de recursieco\"effici\"enten kunnen beschrijven als oplossingen van stelsels die limietgevallen zijn van asymmetrische discrete Painlev\'evergelijkingen. Eens de vergelijkingen zijn opgesteld, hebben we getracht deze te gebruiken om de recursieco\"effici\"enten effectief te berekenen. De na\"ieve aanpak blijkt numeriek instabiel te zijn. Dit probleem hebben we proberen op te lossen door de gezochte recursieco\"effici\"enten te omschrijven als oplossingen van de discrete Painlev\'evergelijkingen die op een bepaalde manier uniek zijn. Hiervoor hebben we een operator nodig die zodanig gekozen is dat de recursieco\"effici\"enten van de semi-klassieke veeltermen overeen komen met een vast punt van de operator. Hoewel we er niet in geslaagd zijn de bewijzen exact te maken, geeft deze aanpak ons wel een stabiele en snelle manier om de recursieco\"effici\"enten te berekenen.1 Introduction 1 1.1 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The orthogonality condition . . . . . . . . . . . . . . . . . . 1 1.1.2 Features of orthogonal polynomials . . . . . . . . . . . . . . 3 1.1.3 The classical orthogonal polynomials on the real line . . . . 6 1.1.4 Classification of (some) classical orthogonal polynomials . . 7 1.1.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 The Painleve equations . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.1 Continuous Painleve equations . . . . . . . . . . . . . . . . 19 1.2.2 Discrete Painleve equations . . . . . . . . . . . . . . . . . . 23 1.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3 Ladder operators for orthogonal polynomials . . . . . . . . . . . . 31 1.3.1 From ladder operators to Lax pairs . . . . . . . . . . . . . . 39 1.4 Outline and goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 Results related to the Askey scheme 45 2.1 Generalized Hermite polynomials and dPI . . . . . . . . . . . . . . 45 2.2 Generalized Charlier polynomials and dPII . . . . . . . . . . . . . . 47 2.3 Generalized Laguerre polynomials . . . . . . . . . . . . . . . . . . 49 2.4 Generalized Meixner polynomials . . . . . . . . . . . . . . . . . . . 52 2.5 Generalized Krawtchouk polynomials . . . . . . . . . . . . . . . . . 53 3 Generalized Laguerre polynomials 55 3.1 A first approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 A second approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Computing the recurrence coefficients . . . . . . . . . . . . . . . . 62 3.4 Beyond the real half line . . . . . . . . . . . . . . . . . . . . . . . . 72 4 Generalized Meixner and Krawtchouk polynomials 75 4.1 Generalized Meixner polynomials . . . . . . . . . . . . . . . . . . . 75 4.2 Generalized Krawtchouk polynomials . . . . . . . . . . . . . . . . . 82 4.3 Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Results related to the q-Askey scheme 89 5.1 Generalized discrete q-Hermite I polynomials and qPI . . . . . . . 89 5.2 Generalized Stieltjes-Wigert polynomials and qPIII . . . . . . . . . 93 5.2.1 Classical extension . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.2 Semiclassical Stieltjes-Wigert polynomials and qPIII . . . . 96 5.3 Generalized q-Laguerre polynomials and qPV . . . . . . . . . . . . 97 5.4 Generalized little q-Laguerre polynomials and qPV . . . . . . . . . 99 6 Discrete q-Hermite I polynomials 101 6.1 The weight function . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.2 The recurrence coefficients . . . . . . . . . . . . . . . . . . . . . . . 104 6.3 Asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.4 Singularity confinement . . . . . . . . . . . . . . . . . . . . . . . . 109 6.5 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 7 Generalized (little) q-Laguerre polynomials 115 7.1 Generalized Stieltjes-Wigert polynomials . . . . . . . . . . . . . . . 116 7.2 Generalized q-Laguerre polynomials . . . . . . . . . . . . . . . . . 121 7.3 Generalized little q-Laguerre polynomials . . . . . . . . . . . . . . 126 8 Conclusion 131 Nederlandstalige samenvatting 133 N.1 Orthogonale veeltermen . . . . . . . . . . . . . . . . . . . . . . . . 133 N.2 Discrete Painlevevergelijkingen . . . . . . . . . . . . . . . . . . . . 135 N.3 Structuur van de tekst . . . . . . . . . . . . . . . . . . . . . . . . . 136 N.4 Eigen resultaten . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 N.4.1 Veralgemeende Laguerreveeltermen . . . . . . . . . . . . . . 137 N.4.2 Veralgemeende Meixner- en Krawtchoukveeltermen . . . . . 138 N.4.3 Veralgemeende discrete q-Hermite I-veeltermen . . . . . . . 140 N.4.4 Veralgemeende Stieltjes-Wigert-, q-Laguerre- en kleine q-Laguerreveeltermen . . . . . . . . . . . . . . . . . . 140 Bibliography 143nrpages: 165status: publishe