Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

Hankel determinants and orthogonal polynomials

We develop a general context for the computation of the determinant of a Hankel matrix H_n = (α_i+j)0i,jn, assuming some suitable conditions for the exponential (or ordinary) generating function of the sequence (α_n)_n0. Several well-known particular cases are thus derived in a unified way.     

On certain estimates for orthogonal polynomials depending on parameters

Рассматривается сис тема ортогональных м ногочленов {P _n (z)} ₀ ^∞ , удовлетворяющ их условиям $$\frac{1}{{2\pi }}\int\limits_0^{2\pi } {P_m (z)\overline {P_n (z)} d\sigma (\theta ) = \left\{ {\begin{array}{*{20}c} {0,m \ne n,P_n (z) = z^n + ...,z = \exp (i\theta ),} \\ {h_n > 0,m = n(n = 0,1,...),} \\ \end{array} } \right.} $$ где σ (θ) — ограниченная неу бывающая на отрезке [0,2π] функция с бесчисленным множе ством точек роста. Вводится последовательность параметров {а_{n 0} ^∞ , независимых дру г от друга и подчиненных единств енному ограничению { ¦а_n¦<1} ₀ ^∞ ; все многочлены {Р _n (z)} 0/∞ можно найти по формуле $$P_0 = 1,P_{k + 1(z)} = zP_k (z) - a_k P_k^ * (z),P_k^ * (z) = z^k \bar P_k \left( {\frac{1}{z}} \right)(k = 0,1,...)$$ . Многие свойства и оце нки для {P _n (z)} ₀ ^∞ и (θ) можн о найти в зависимости от этих параметров; например, условие \(\mathop \Sigma \limits_{n = 0}^\infty \left| {a_n } \right|^2< \infty \) , бо лее общее, чем условие Г. Cerë, необходимо и достато чно для справедливости а симптотической форм улы в области ¦z¦>1. Пользуясь этим ме тодом, можно найти также реш ение задачи В. А. Стекло ва.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Sobolev orthogonal polynomials in computing of Hankel determinants

In this paper, we study closed form evaluation for some special Hankel determinants arising in combinatorial analysis, especially for the bidirectional number sequences. We show that such problems are directly connected with the theory of quasi-definite discrete Sobolev orthogonal polynomials. It opens a lot of procedural dilemmas which we will try to exceed. A few examples deal with Fibonacci numbers and power sequences will illustrate our considerations. We believe that our usage of Sobolev orthogonal polynomials in Hankel determinant computation is quite new.     

On classical orthogonal polynomials

Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.     

The variation of zeros of certain orthogonal polynomials

We show that the largest zero of a birth and death process polynomial increases (decreases) with a parameter ν if the birth rates and death rates are increasing (decreasing) functions of ν. A similar result is proved for the smallest zero of a birth and death process polynomial. These results are applicable to several sets of orthogonal polynomials. We show that the largest zero of a random walk polynomial is a monotone function of a parameter ν if certain coefficients related to the birth rates and the death rates are monotone functions of ν. We prove that if x_ν is a positive zero of a Lommel polynomial h_n,ν(x), ν > 0, then as ν increases x_ν will decrease but νx_ν will increase. Limiting cases of these results imply known facts concerning positive zeros of Bessel functions. We also establish similar results for a general class of discrete orthogonal polynomials.     

On separating systems whose elements are sets of at most k elements

A system A₁,…,A_m of subsets of X?{1,…,n} is called a separating system if for any two distinct elements of X there is a set A_i (1?i?m) that contains exactly one of the two elements. We investigate separating systems where each set A_i has at most k elements and we are looking for minimal separating systems, that means separating systems with the least number of subsets. We call this least number m(n,k). Katona has proved good bounds on m(n,k) but his proof is very complicated. We give a shorter and easier proof. In doing so we slightly improve the upper bound of Katona.     

On expansions in orthogonal polynomials

A recently introduced fast algorithm for the computation of the first N terms in an expansion of an analytic function into ultraspherical polynomials consists of three steps: Firstly, each expansion coefficient is represented as a linear combination of derivatives; secondly, it is represented, using the Cauchy integral formula, as a contour integral of the function multiplied by a kernel; finally, the integrand is transformed to accelerate the convergence of the Taylor expansion of the kernel, allowing for rapid computation using Fast Fourier Transform. In the current paper we demonstrate that the first two steps remain valid in the general setting of orthogonal polynomials on the real line with finite support, orthogonal polynomials on the unit circle and Laurent orthogonal polynomials on the unit circle.     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.