Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients, let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For a canonical Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0, and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for canonical Geronimus transformations.
Tensor exponential function is an important function that is widely used. In this paper, tensor Pade´-type approximant (TPTA) is defined by introducing a generalized linear functional for the first time. The expression of TPTA is provided with the generating function form. Moreover, by means of formal orthogonal polynomials, we propose an efficient algorithm for computing TPTA. As an application, the TPTA for computing the tensor exponential function is presented. Numerical examples are given to demonstrate the efficiency of the proposed algorithm.
A pair(𝒰,𝒱)of Hermitian regular linear functionals on the unit circle is said to be a(1,1)-coherent pair if their corresponding sequences of monic orthogonal polynomials{ϕn(x)}n≥0and{ψn(x)}n≥0satisfyϕn[1](z)+anϕn-1[1](z)=ψn(z)+bnψn-1(z),an≠0,n≥1, whereϕn[1](z)=ϕn+1′(z)/(n+1). In this contribution, we consider the cases when𝒰is the linear functional associated with the Lebesgue and Bernstein-Szegő measures, respectively, and we obtain a classification of the situations where𝒱is associated with either a positive nontrivial measure or its rational spectral transformation.
This is a continuation of our previous investigations on polynomials orthogonal with respect to the linear functional L : P?C, where L = ?1 -1 p(x) d?(x), d?(x) = (1-x?)?-1/2 exp(i?x) dx, and P is a linear space of all algebraic polynomials. Here, we prove an extension of our previous existence theorem for rational ? ? (-1/2,0], give some hypothesis on three-term recurrence coefficients, and derive some differential relations for our orthogonal polynomials, including the second order differential equation.
Given a quasi-definite linear functional u in the linear space of polynomials with complex coefficients let us consider the corresponding sequence of monic orthogonal polynomials (SMOP in short) (Pn)n≥0. For the Christoffel transformation u˜=(x−c)u with SMOP (P˜n)n≥0, we are interested to study the relation between u˜ and u(1)˜, where u(1) is the linear functional for the associated orthogonal polynomials of the first kind (Pn(1))n≥0 and u(1)˜=(x−c)u(1) is its Christoffel transformation. This problem is also studied for the Geronimus transformations.
We give the system of Laguerre-Freud equations for the recurrence coefficients
?n, ?n+1, n ? 0 of orthogonal polynomials with respect to a Hq-semiclassical
form (linear functional) of class one. The system is solved in the case when
?n = tn-1-tn and n+1 =-t2n with tn?0, n ? 0 and t-1 = 0. There are
essentially three canonical cases.
Abstract
We study properties of the form (linear functional) u = λ(x – a)–1
ν + δb
, where ν is a regular form. We give a necessary and sufficient condition for the regularity of the form u. The coefficients of a three-term recurrence relation, satisfied by the corresponding sequence of orthogonal polynomials, are given explicitly. The semi-classical character of the founded families is studied. We conclude by giving some examples.
We study the three-term recurrence coefficients [Formula: see text] of polynomial sequences orthogonal with respect to a perturbed linear functional depending on a variable [Formula: see text] We obtain power series expansions in [Formula: see text] and asymptotic expansions as [Formula: see text] We use our results to settle some conjectures proposed by Walter Van Assche and collaborators.
AbstractWe give the system of Laguerre–Freud equations for the recurrence coefficients {\beta_{n}}, {\gamma_{n+1}}, {n\geq 0}, of orthogonal polynomials with respect to a D-Laguerre–Hahn form (linear functional) of class one. The system is solved in the case when {\beta_{0}=-\theta_{0}}, {\beta_{n+1}=\theta_{n}-\theta_{n+1}} and {\gamma_{n+1}=-\theta_{n}^{2}} with {\theta_{n}\neq 0}, {n\geq 0}. There are essentially three canonical cases.
Given a sequence of monic orthogonal polynomials (MOPS),{Pn}, with respect to a quasi-definite linear functionalu, we find necessary and sufficient conditions on the parametersanandbnfor the sequencePn(x)+anPn−1(x)+bnPn−2(x), n≥1P0(x)=1,P−1(x)=0to be orthogonal. In particular, we can find explicitly the linear functionalvsuch that the new sequence is the corresponding family of orthogonal polynomials. Some applications for Hermite and Tchebychev orthogonal polynomials of second kind are obtained.We also solve a problem of this type for orthogonal polynomials with respect to a Hermitian linear functional.
Spectral Transformations and Associated Linear Functionals of the First Kind