Our primary result concerns the positivity of specific kernels constructed using the q-ultraspherical polynomials. In other words, it concerns a two-parameter family of bivariate, compactly supported distributions. Moreover, this family has a property that all its conditional moments are polynomials in the conditioning random variable. The significance of this result is evident for individuals working on distribution theory, orthogonal polynomials, q-series theory, and the so-called quantum polynomials. Therefore, it may have a limited number of interested researchers. That is why, we put our results into a broader context. We recall the theory of Hilbert–Schmidt operators and the idea of Lancaster expansions (LEs) of the bivariate distributions absolutely continuous with respect to the product of their marginal distributions. Applications of LE can be found in Mathematical Statistics or the creation of Markov processes with polynomial conditional moments (the most well-known of these processes is the famous Wiener process).

We calculate the moments of the characteristic polynomials of $N\times N$ matrices drawn from the Hermitian ensembles of Random Matrix Theory, at a position t in the bulk of the spectrum, as a series expansion in powers of t. We focus in particular on the Gaussian Unitary Ensemble. We employ a novel approach to calculate the coefficients in this series expansion of the moments, appropriately scaled. These coefficients are polynomials in N. They therefore grow as $N\to\infty$ , meaning that in this limit the radius of convergence of the series expansion tends to zero. This is related to oscillations as t varies that are increasingly rapid as N grows. We show that the $N\to\infty$ asymptotics of the moments can be derived from this expansion when $t=0$ . When $t\ne 0$ we observe a surprising cancellation when the expansion coefficients for N and $N+1$ are formally averaged: this procedure removes all of the N-dependent terms leading to values that coincide with those expected on the basis of previously established asymptotic formulae for the moments. We obtain as well formulae for the expectation values of products of the secular coefficients.

We consider the spectral analysis of several examples of bilateral birth–death processes and compute explicitly the spectral matrix and the corresponding orthogonal polynomials. We also use the spectral representation to study some probabilistic properties of the processes, such as recurrence, the invariant distribution (if it exists), and the probability current.

$\bar {\partial } $ -extension of the matrix Riemann–Hilbert method is used to study asymptotics of the polynomials $ P_n(z) $ satisfying orthogonality relations

$$ \begin{align*} \int_{-1}^1 x^lP_n(x)\frac{\rho(x)dx}{\sqrt{1-x^2}}=0, \quad l\in\{0,\ldots,n-1\}, \end{align*} $$

where $ \rho (x) $ is a positive $ m $ times continuously differentiable function on $ [-1,1] $ , $ m\geq 3 $ .

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

The line connecting rare earth elements (REE) in chondrite-normalised plots can be represented by a smooth polynomial function using λ shape coefficients as described by O'Neill (2016). In this study, computationally generated λ combinations are used to construct artificial chondrite-normalised REE patterns that encompass most REE patterns likely to occur in natural materials. The dominant REE per pattern is identified, which would lead to its inclusion in a hypothetical mineral suffix, had this mineral contained essential REE. Furthermore, negative Ce and Y anomalies, common in natural minerals, are considered in the modelled REE patterns to investigate the effect of their exclusion on the relative abundance of the remainder REE. The dominant REE in a mineral results from distinct pattern shapes requiring specific fractionation processes, thus providing information on its genesis. Minerals dominated by heavy lanthanides are rare or non-existent, even though the present analysis shows that REE patterns dominated by Gd, Dy, Er and Yb are geologically plausible. This discrepancy is caused by the inclusion of Y, which dominates heavy REE budgets, in mineral name suffixes. The focus on Y obscures heavy lanthanide mineral diversity and can lead to various fractionation processes to be overlooked. Samarium dominant minerals are known, even though deemed unlikely by the computational model, suggesting additional fractionation processes that are not well described by λ shape coefficients. Positive Eu anomalies only need to be moderate in minerals depleted in the light REE for Eu to be the dominant REE, thus identifying candidate rocks in which the first Eu dominant mineral might be found. Here, I present an online tool, called ALambdaR that allows interactive control of λ shape coefficients and visualisation of resulting REE patterns.

Outflow from a young star might be regarded as approximately equivalent to flow from a point source. If the fluid consists of charged particles, then the magnetic fields produced are governed by Faraday's law. This simple first approximation yields a linear partial differential equation in spherical polar coordinates, and its solution may be represented as the product of a Legendre polynomial with some function of the radial coordinate. This radial function is shown to involve orthogonal polynomials. Their properties are investigated and recurrence formulae for them are derived. Some of the magnetic fields generated by this simple model are illustrated.

We show that if $v$ is a regular semi-classical form (linear functional), then the symmetric form $u$ defined by the relation ${{x}^{2}}\sigma u\,=\,-\lambda v$, where $\left( \sigma f \right)\left( x \right)\,=\,f\left( {{x}^{2}} \right)$ and the odd moments of $u$ are 0, is also regular and semi-classical form for every complex $\lambda $ except for a discrete set of numbers depending on $v$. We give explicitly the three-term recurrence relation and the structure relation coefficients of the orthogonal polynomials sequence associated with $u$ and the class of the form $u$ knowing that of $v$. We conclude with an illustrative example.

Predictability of revenue and costs to both operators and users is critical for payment schemes. We study the issue of the design of payment schemes in networks with bandwidth sharing. The model we consider is a processor sharing system that is accessed by various classes of users with different processing requirements or file sizes. The users are charged according to a Vickrey–Clarke–Groves mechanism because of its efficiency and fairness when logarithmic utility functions are involved. Subject to a given mean revenue for the operator, we study whether it is preferable for a user to pay upon arrival, depending on the congestion level, or whether the user should opt to pay at the end. This leads to a study of the volatility of payment schemes and we show that opting for prepayment is preferable from a user point of view. The analysis yields new results on the asymptotic behavior of conditional response times for processor sharing systems and connections to associated orthogonal polynomials.

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

The finite Fourier transform of a family of orthogonal polynomials is the usual transform of these polynomials extended by $0$ outside their natural domain of orthogonality. Explicit expressions are given for the Legendre, Jacobi, Gegenbauer and Chebyshev families.

The design optimization and analysis of charged particle beam systems employing intense beams requires a robust and accurate Poisson solver. This paper presents a new type of Poisson solver which allows the effects of space charge to be elegantly included into the system dynamics. This is done by casting the charge distribution function into a series of basis functions, which are then integrated with an appropriate Green's function to find a Taylor series of the potential at a given point within the desired distribution region. In order to avoid singularities, a Duffy transformation is applied, which allows singularity-free integration and maximized convergence region when performed with the help of Differential Algebraic methods. The method is shown to perform well on the examples studied. Practical implementation choices and some of their limitations are also explored.

In the present paper we study the reconstruction of a structured quadratic pencil fromeigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrixpolynomial

QP(λ) = M λ2+Cλ +K,

where M,C, andK are realsquare matrices. The approach developed in the paper is based on the theory of orthogonalpolynomials on the real line. The results can be applied to more general distribution ofeigenvalues. The problem with added single eigenvector is also briefly discussed. As anillustration of the reconstruction method, the eigenvalue problem on linearized stabilityof certain class of stationary exact solution of the Navier-Stokes equations describingatmospheric flows on a spherical surface is reformulated as a simple mass-spring system bymeans of this method.