Nonnegative and strictly positive linearization of Jacobi and generalized Chebyshev polynomials

In the theory of orthogonal polynomials, as well as in its intersection with harmonic analysis, it is an important problem to decide whether a given orthogonal polynomial sequence $(P_n(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products, i.e., the product of any two $P_m(x),P_n(x)$ is a conical combination of the polynomials $P_{|m-n|}(x),\ldots,P_{m+n}(x)$. Since the coefficients in the arising expansions are often of cumbersome structure or not explicitly available, such considerations are generally very nontrivial. In 1970, G. Gasper was able to determine the set $V$ of all pairs $(\alpha,\beta)\in(-1,\infty)^2$ for which the corresponding Jacobi polynomials $(R_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, normalized by $R_n^{(\alpha,\beta)}(1)\equiv1$, satisfy nonnegative linearization of products. In 2005, R. Szwarc asked to solve the analogous problem for the generalized Chebyshev polynomials $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$, which are the quadratic transformations of the Jacobi polynomials and orthogonal w.r.t. the measure $(1-x^2)^{\alpha}|x|^{2\beta+1}\chi_{(-1,1)}(x)\,\mathrm{d}x$. In this paper, we give the solution and show that $(T_n^{(\alpha,\beta)}(x))_{n\in\mathbb{N}_0}$ satisfies nonnegative linearization of products if and only if $(\alpha,\beta)\in V$, so the generalized Chebyshev polynomials share this property with the Jacobi polynomials. Moreover, we reconsider the Jacobi polynomials themselves, simplify Gasper's original proof and characterize strict positivity of the linearization coefficients. Our results can also be regarded as sharpenings of Gasper's one.Comment: The second version puts more emphasis on strictly positive linearization of Jacobi polynomials. We reorganized the structure, added several references and corrected a few typos. We added a geometric interpretation of the set $V^{\prime}$ and some comments on its interior. We added a detailed comparison to Gasper's classical result. Title and abstract were changed. These are the main change

Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Tur\'{a}n type inequalities and transformations into more convenient orthogonal polynomial systems.Comment: Main changes towards first version: The part on associated symmetric Pollaczek polynomials was extended (with more emphasis on Tur\'{a}n's inequality and including a larger parameter region), and the part on little $q$-Legendre polynomials became a separate paper. We added several references and corrected a few typos. Title, abstract and MSC class were change

Harmonic analysis of little qq-Legendre polynomials

Many classes of orthogonal polynomials satisfy a specific linearization property giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as $L^1$-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual behavior strongly depends on these underlying polynomials. We study the little $q$-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their $L^1$-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the $L^1$-algebras associated with the little $q$-Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these $L^1$-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner derivation), which is known to be shared by any $L^1$-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior.Comment: The paper is essentially also a part of the first version of arXiv:1806.00339 [math.FA]. It is now a separate paper because the associated symmetric Pollaczek part of arXiv:1806.00339 [math.FA] was extended. Compared to the (first version of) arXiv:1806.00339, we extended and added some results on little $q$-Legendre polynomials, modified the notation and added some graphic

Expansions and characterizations of sieved random walk polynomials

We consider random walk polynomial sequences $(P_n(x))_{n\in\mathbb{N}_0}\subseteq\mathbb{R}[x]$ given by recurrence relations $P_0(x)=1$, $P_1(x)=x$, $x P_n(x)=(1-c_n)P_{n+1}(x)+c_n P_{n-1}(x)\;(n\in\mathbb{N})$ with $(c_n)_{n\in\mathbb{N}}\subseteq(0,1)$. For every $k\in\mathbb{N}$, the $k$-sieved polynomials $(P_n(x;k))_{n\in\mathbb{N}_0}$ arise from the recurrence coefficients $c(n;k):=c_{n/k}$ if $k|n$ and $c(n;k):=1/2$ otherwise. A main objective of this paper is to study expansions in the Chebyshev basis $\{T_n(x):n\in\mathbb{N}_0\}$. As an application, we obtain explicit expansions for the sieved ultraspherical polynomials. Moreover, we introduce and study a sieved version $\mathrm{D}_k$ of the Askey-Wilson operator $\mathcal{D}_q$. It is motivated by the sieved ultraspherical polynomials, a generalization of the classical derivative and obtained from $\mathcal{D}_q$ by letting $q$ approach a $k$-th root of unity. However, for $k\geq2$ the new operator $\mathrm{D}_k$ on $\mathbb{R}[x]$ has an infinite-dimensional kernel (in contrast to its ancestor), which leads to additional degrees of freedom and characterization results for $k$-sieved random walk polynomials. Similar characterizations are obtained for a sieved averaging operator $\mathrm{A}_k$

Genomic Dissection of Bipolar Disorder and Schizophrenia, Including 28 Subphenotypes

publisher: Elsevier articletitle: Genomic Dissection of Bipolar Disorder and Schizophrenia, Including 28 Subphenotypes journaltitle: Cell articlelink: https://doi.org/10.1016/j.cell.2018.05.046 content_type: article copyright: © 2018 Elsevier Inc

Dydrogesterone as an oral alternative to vaginal progesterone for IVF luteal phase support: A systematic review and individual participant data meta-analysis.

The aim of this systematic review and meta-analysis was to conduct a comprehensive assessment of the evidence on the efficacy and safety of oral dydrogesterone versus micronized vaginal progesterone (MVP) for luteal phase support. Embase and MEDLINE were searched for studies that evaluated the effect of luteal phase support with daily administration of oral dydrogesterone (20 to 40 mg) versus MVP capsules (600 to 800 mg) or gel (90 mg) on pregnancy or live birth rates in women undergoing fresh-cycle IVF (protocol registered at PROSPERO [CRD42018105949]). Individual participant data (IPD) were extracted for the primary analysis where available and aggregate data were extracted for the secondary analysis. Nine studies were eligible for inclusion; two studies had suitable IPD (full analysis sample: n = 1957). In the meta-analysis of IPD, oral dydrogesterone was associated with a significantly higher chance of ongoing pregnancy at 12 weeks of gestation (odds ratio [OR], 1.32; 95% confidence interval [CI], 1.08 to 1.61; P = 0.0075) and live birth (OR, 1.28; 95% CI, 1.04 to 1.57; P = 0.0214) compared to MVP. A meta-analysis combining IPD and aggregate data for all nine studies also demonstrated a statistically significant difference between oral dydrogesterone and MVP (pregnancy: OR, 1.16; 95% CI, 1.01 to 1.34; P = 0.04; live birth: OR, 1.19; 95% CI, 1.03 to 1.38; P = 0.02). Safety parameters were similar between the two groups. Collectively, this study indicates that a higher pregnancy rate and live birth rate may be obtained in women receiving oral dydrogesterone versus MVP for luteal phase support

Medication wrong route administration: a poisons center-based study

OBJECTIVES: To describe clinical effects, circumstances of occurrence, management and outcomes of cases of inadvertent administration of medications by an incorrect parenteral route. METHODS: Retrospective single-center consecutive review of parenteral route errors of medications, reported to our center between January 2006 and June 2010. We collected demographic data and information on medications, route and time of administration, severity of symptoms/signs, treatment, and outcome. RESULTS: Seventy-eight cases (68 adults, 10 children) were available for analysis. The following wrong administration routes were recorded: paravenous (51%), intravenous (33%), subcutaneous (8%), and others (8%). Medications most frequently involved were iodinated x-ray contrast media (11%) and iron infusions (9%). Twenty-eight percent of the patients were asymptomatic and 54% showed mild symptoms; moderate and severe symptoms were observed in 9% and 7.7%, respectively, and were mostly due to intravenous administration errors. There was no fatal outcome. In most symptomatic cases local nonspecific treatment was performed. CONCLUSIONS: Enquiries concerning administration of medicines by an incorrect parenteral route were rare, and mainly involved iodinated x-ray contrast media and iron infusions. Most events occurred in adults and showed a benign clinical course. Although the majority of exposures concerned the paravenous route, the occasional severe cases were observed mainly after inadvertent intravenous administration