Determination of All Coherent Pairs

Abstract

A pair of quasi-definite linear functionals [u 0 , u 1 ] on the set of polynomials is called a coherent pair if their corresponding sequences of monic orthogonal polynomials [P n ] and [T n ] satisfy a relation

A matrix approach for the semiclassical and coherent orthogonal polynomials

“…In [19], a complete classification of ð1; 0Þ coherent pairs of regular linear functionals was given. However, the ð1; 0Þ coherent pairs have been also studied in [1,12 14].…”

“…From the point of view of appli cations, the potential interest of such Sobolev orthogonal polynomials appears when you consider spectral (Galerkin and collocation) methods for boundary value problems associated with Schrödinger equations whose potentials are related with such coherent pairs. In 1997, in [19], Meijer determined all (1, 0) coherent pairs ðU; VÞ of regular linear functionals. He proved that at least one of the linear functionals (U or V) must be classical (Laguerre or Jacobi).…”

(M,N)-Coherent pairs of linear functionals and Jacobi matrices

“…Besides, when m 1; ðU; VÞ is said to be an ðM; NÞ coherent pair. The above definition unifies the generalizations given in the literature of the concept of coherent pair (in our terminology, ð1; 0Þ coherent pair) introduced by Iserles, Koch, Nørsett, and Sanz Serna in [8] (see also the work of Meijer [19]). In fact, Delgado and Marcellán [5] introduced ð1; 1Þ coherent pairs, Kwon, Lee, and Marcellán [12] considered ð2; 0Þ coherent pairs, Maroni and Sfaxi [17,18] looked at the ðM þ N; 2M þ 1Þ coherence relation, Marcellán, Martínez Finkelshtein, and Moreno Balcázar [14] presented the ðM; 0Þ coherence relation, Alfaro, Marcellán, Peña, and Rezola [1,2] studied ð1; 1Þ coherent pairs of order 0, Petronilho [20] considered the ðM; NÞ coherence relation of order 0, de Jesus and Petronilho [10,11] analyzed ðM; NÞ coherent pairs, Branquinho and Rebocho [3] looked at the ð1; 0Þ coherence relation of order 2, and, Marcellán and Pinzón Cortés [15] studied ð1; 1Þ coherent pairs of order m. For a review about these and other related works, see for instance, the introductory sections in the papers [15,11].…”

“…Let us consider the particular case of ð1; 0Þ coherence studied by Meijer in [19]. In this case, the matrices A 1 ; A, and B in (3.6) and (3.7) are, respectively, a nonsingular lower bidiagonal matrix (1=ðn þ 1Þ are the entries of its main diag onal), a nonsingular upper bidiagonal matrix (a 1;n =n are the entries of its main diagonal), and the identity matrix.…”

Companion Linear Functionals and Sobolev Inner Products: A Case Study

“…In [14] the description of all coherent pairs of measures is done. As a conclusion, if (µ 0 , µ 1 ) is a pair of coherent measures, i.e., the corresponding sequences {R n } and {P n } of monic orthogonal polynomials are related by…”