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This paper is concerned with double sequencesC={Cn}n=−∞/∞ of Hermitian matrices with complex entriesCn∈Ms×s) and formal Laurent seriesL0(z)=−Σk=1∞C−kzk andL∞(z)=Σk=0∞Ckz−k. Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Pade approximants to the pair (L0,L∞) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.
49Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems
Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems
47On a class of matrix orthogonal polynomials on the real line
On a class of matrix orthogonal polynomials on the real line
90Orthogonal polynomials and some q-beta integrals of Ramanujan
Orthogonal polynomials and some q-beta integrals of Ramanujan
18On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials
On matrix Hurwitz type polynomials and their interrelations to Stieltjes positive definite sequences and orthogonal matrix polynomials
Read more197Orthogonal matrix polynomials and higher-order recurrence relations
Orthogonal matrix polynomials and higher-order recurrence relations
9Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs
Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs
11Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle
Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle
25Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice
Matrix valued Hermite polynomials, Burchnall formulas and non-abelian Toda lattice
13Differential coefficients of orthogonal matrix polynomials
Differential coefficients of orthogonal matrix polynomials
Matrix Bessel biorthogonal polynomials: a Riemann–Hilbert approach
We consider matrix orthogonal polynomials related to Bessel type matrices of weights that can be defined in terms of a given matrix Pearson equation. From a Riemann–Hilbert problem we derive first and second order differential relations for the matrix orthogonal polynomials and functions of second kind. It is shown that the corresponding matrix recurrence coefficients satisfy a non-Abelian extension of a family of discrete Painlevé d-PIV equations. We present some nontrivial examples of matrix orthogonal polynomials of Bessel type.
Read more26Darboux transformations for CMV matrices
Darboux transformations for CMV matrices
Orthogonality measures for orthogonal matrix polynomials with periodic coefficients of recurrence relations
where P−1(t) = θ, P0(t) ∈ CN×N , the An are nondegenerate matrices, the Bn are Hermitian matrices, and θ denotes the zero matrix. By the matrix analog of the sequence of Chebyshev polynomials of the second kind [3] we mean the sequence of orthonormal matrix polynomials U n defined by the recurrence relations (1) with An = A, Bn = B, and the initial conditions U −1 (t) = θ, U A,B 0 (t) = I. The measure WA,B(x), with respect to which the polynomials U n are orthonormal was obtained in [1]. In the case of a positive definite matrix A, the measureWA,B(x) has the following form:
Read more39The extended Krylov subspace method and orthogonal Laurent polynomials
The extended Krylov subspace method and orthogonal Laurent polynomials
4On 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.
Read more2On Functions of the Second Kind in Orthogonal Polynomial Theory
A given weight function \(p(x)\) on an interval \([a, b]\) defines uniquely, subject to normalization, a sequence of orthogonal polynomials \(P_n(x)\) and their corresponding sequence of functions of the second kind $$\begin{aligned} Q_n(x)=\int _a^b\frac{P_n(t)p(t)dt}{x-t},\quad \ x\in \mathbf{C } \backslash [a,b]. \end{aligned}$$ corresponding to them. This paper focuses on the reverse problem of characterizing orthogonal polynomials by means of functions of the second kind, together with the properties of such functions of the second kind. Functions of the second kind for orthogonal polynomials are also of particular interest in that they differ only slightly from the second solution of the differential equation satisfied by the orthogonal polynomials.
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