Related papers: Sobolev Freud polynomials
A Littlewood polynomial is a single-variable polynomial all of whose coefficients lie in $\{ \pm 1\}$. We establish the leading term asymptotics of the number of reciprocal or skew-reciprocal Littlewood polynomials with square discriminant.…
We study the zero distribution of the sum of the first $n$ polynomials satisfying a three-term recurrence whose coefficients are linear polynomials. We also extend this sum to a linear combination, whose coefficients are powers of $az+b$…
We introduce a new representation for the rescaled Appell polynomials and use it to obtain asymptotic expansions to arbitrary order. This representation consists of a finite sum and an integral over a universal contour (i.e. independent of…
Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2}, >...c,z^{-k},z^{k},...c \rbrace$ with respect to the real inner product $(f,g) \mapsto \int_{\mathbb{R}}f(s)g(s) \exp (-\mathscr{N} V(s)) \md s$, $\mathscr{N} \in…
We investigate generalizations of the Charlier polynomials on the lattice $\mathbb{N}$, on the shifted lattice $\mathbb{N}+1-\beta$ and on the bi-lattice $\mathbb{N}\cup (\mathbb{N}+1-\beta)$. We show that the coefficients of the three-term…
In the context of orthogonal polynomials in the plane we introduce the notion of a polynomially small (PS) perturbation of a measure. In such a case we establish relative asymptotic results for the two sequences of the associated…
In this paper we study sequences of vector orthogonal polynomials. The vector orthogonality presented here provides a reinterpretation of what is known in the literature as matrix orthogonality. These systems of orthogonal polynomials…
We obtain explicit upper and lower bounds on the norms of the spectral projections of the non-self-adjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials.
We study the dependence of recurrence coefficients in the three-term recurrence relation for orthogonal polynomials with a certain deformation of the $q$-Laguerre weight on the degree parameter $n$. We show that this dependence is described…
We study the asymptotic behavior of Multiple Meixner polynomials of first and second kind, respectively (see J. Arves\'u et al. J. Comput. Appl. Math., 153, (2003)). We use an algebraic function formulation for the solution of the…
By using a generalization of Sturm-Liouville problems in $q$-difference spaces, a class of symmetric $q$-orthogonal polynomials with four free parameters is introduced. The standard properties of these polynomials, such as a second order…
Iterated Geronimus transformations generate Sobolev-type orthogonal polynomials from classical families. We establish a direct equivalence between a Sobolev inner product involving point evaluation and the first derivative at a point a…
In this paper, we study the sequence of orthogonal polynomials $\{S_n\}_{n=0}^{\infty}$ with respect to the Sobolev-type inner product $$\langle f,g \rangle= \int_{-1}^{1} f(x) g(x) \,d\mu(x) +\sum_{j=1}^{N} \eta_{j} \,f^{(d_j)}(c_{j})…
We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic…
The 3-term recurrence relation for Hermite polynomials was recently generalized to a recurrence relation for Wronskians of Hermite polynomials. In this note, a similar generalization for Laguerre polynomials is obtained.
We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a…
We consider orthogonal polynomials $\{p_{n,N}(x)\}_{n=0}^{\infty}$ on the real line with respect to a weight $w(x)=e^{-NV(x)}$ and in particular the asymptotic behaviour of the coefficients $a_{n,N}$ and $b_{n,N}$ in the three term…
We propose a theory of combinatorially explicit Schubert polynomials which represent the Schubert classes in the Borel presentation of the cohomology ring of orthogonal flag varieties. We use these polynomials to describe the arithmetic…
Let the Sobolev-type inner product <f,g> = \int fg d mu_0+ int f' g' d mu_1 with mu_0 = w + M delta_c, mu_1= N delta_c where w is the Jacobi weight, c is either 1 or -1 and M, N >= 0. We obtain estimates and asymptotic properties on [-1,1]…
For every system $\{ p_n(z) \}_{n=0}^\infty$ of OPRL or OPUC, we construct Sobolev orthogonal polynomials $y_n(z)$, with explicit integral representations involving $p_n$. Two concrete families of Sobolev orthogonal polynomials (depending…