J. Petronilho

Classical orthogonal polynomials: A functional approach

We characterize the so-called classical orthogonal polynomials (Hermite, Laguerre, Jacobi, and Bessel) using the distributional differential equation D(φu)=ψu. This result is naturally not new. However, other characterizations of classical orthogonal polynomials can be obtained more easily from this approach. Moreover, three new properties are obtained.

Explicit inverses of some tridiagonal matrices

Abstract We give explicit inverses of tridiagonal 2-Toeplitz and 3-Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix.

Eigenproblems for tridiagonal 2-Toeplitz matrices and quadratic polynomial mappings☆

Abstract Given a system of monic orthogonal polynomials (MOPS) { P n ( x )} n ⩾ 0 , we characterize all the sequences of monic orthogonal polynomials { Q n ( x )} n ⩾ 0 such that Q 1 ( x ) = x − b , Q 2 n ( x ) = P n [ π 2 ( x )], n = 0, 1, 2, …, where π 2 is a fixed polynomial of degree exactly 2 and b is a fixed complex number. With an appropriate choice of the MOPS { P n ( x )} n ⩾ 0 , our results enables us to solve the eigenproblem of a tridiagonal 2-Toeplitz matrix, giving an alternative proof to a recent result by M. J. C. Gover. We also find the relations between the Jacobi matrices corresponding to the MOPS { P n ( x )} n ⩾ 0 and { Q n ( x )} n ⩾ 0 . Finally, we show that if { P n ( x )} n ⩾ 0 is a semiclassical orthogonal polynomial sequence, then so is { Q n ( x )} n ⩾ 0 , and, in particular, we analyze the classical case in detail.

WKB Approximation and Krall-Type Orthogonal Polynomials

We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous one‘perturbed’ by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied.

On inverse problems for orthogonal polynomials, I

Abstract Bonan et al. (1987) gave an apparent generalization of semiclassical orthogonal polynomial sequences for positive measures as an inverse problem for orthogonal polynomials. We study a more general situation for regular orthogonal polynomials. The connection between the corresponding linear functions is obtained. The basic result is the semiclassical character of such functionals.

Path polynomials of a circuit: a constructive approach

Let Pk denote the polynomial of the path on k vertices. We describe completely the matrix Pk (Cn ), where Cn is the circuit on n vertices, using some important concepts of theory of circulant matrices. We also consider Q k , the polynomial of the circuit on kvertices. Using orthogonal polynomials we present constructive proofs of some results obtained recently by Bapat and Lai, Beezer and Ronghua.

Orthogonal polynomials generated by a linear structure relation: Inverse problem

Abstract Let ( P n ) n and ( Q n ) n be two sequences of monic polynomials linked by a type structure relation such as Q n ( x ) + r n Q n − 1 ( x ) = P n ( x ) + s n P n − 1 ( x ) + t n P n − 2 ( x ) , where ( r n ) n , ( s n ) n and ( t n ) n are sequences of complex numbers. First, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences ( P n ) n and ( Q n ) n are orthogonal with respect to regular moment linear functionals u and v , respectively. Second, assuming that the above relation is non-degenerate and ( P n ) n is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence ( Q n ) n in terms of the coefficients of the polynomials Φ and Ψ which appear in the rational transformation (in the distributional sense) Φ u = Ψ v . Some illustrative examples of the developed theory are presented.

On linearly related sequences of difference derivatives of discrete orthogonal polynomials

Let ? be either ω ? C ? { 0 } or q ? C ? { 0 , 1 } , and let D ? be the corresponding difference operator defined in the usual way either by D ω p ( x ) = p ( x + ω ) - p ( x ) ω or D q p ( x ) = p ( q x ) - p ( x ) ( q - 1 ) x . Let U and V be two moment regular linear functionals and let { P n ( x ) } n ? 0 and { Q n ( x ) } n ? 0 be their corresponding orthogonal polynomial sequences (OPS). We discuss an inverse problem in the theory of discrete orthogonal polynomials involving the two OPS { P n ( x ) } n ? 0 and { Q n ( x ) } n ? 0 assuming that their difference derivatives D ? of higher orders m and k (resp.) are connected by a linear algebraic structure relation such as ? i = 0 M a i , n D ? m P n + m - i ( x ) = ? i = 0 N b i , n D ? k Q n + k - i ( x ) , n ? 0 , where M , N , m , k ? N ? { 0 } , a M , n ? 0 for n ? M , b N , n ? 0 for n ? N , and a i , n = b i , n = 0 for i n . Under certain conditions, we prove that U and V are related by a rational factor (in the ? - distributional sense). Moreover, when m ? k then both U and V are D ? -semiclassical functionals. This leads us to the concept of ( M , N ) - D ? -coherent pair of order ( m , k ) extending to the discrete case several previous works. As an application we consider the OPS with respect to the following Sobolev-type inner product { p ( x ) , r ( x ) } λ , ? = { U , p ( x ) r ( x ) } + λ { V , ( D ? m p ) ( x ) ( D ? m r ) ( x ) } , λ 0 , assuming that U and V (which, eventually, may be represented by discrete measures supported either on a uniform lattice if ? = ω , or on a q -lattice if ? = q ) constitute a ( M , N ) - D ? -coherent pair of order m (that is, an ( M , N ) - D ? -coherent pair of order ( m , 0 ) ), m ? N being fixed.

A note on the spectra of tridiagonal matrices

Using orthogonal polynomials, we give a different approach to some recent results on tridiagonal matrices.

Spectra of certain Jacobi operators

We discuss the spectra and, in particular, the essential spectra, of some bounded self-adjoint Jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings.