共查询到20条相似文献,搜索用时 15 毫秒
1.
Second order partial differential equations for gradients of orthogonal polynomials in two variables
Lidia Fernández Teresa E. Pérez Miguel A. Piñar 《Journal of Computational and Applied Mathematics》2007
In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one. 相似文献
2.
Mourad E.H. Ismail Plamen Simeonov 《Journal of Mathematical Analysis and Applications》2011,376(1):259-274
We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations. 相似文献
3.
André Draux 《Numerical Algorithms》1996,11(1):143-158
From some results concerning the formal orthogonal polynomials, already proved in [5], we develop new properties of generalized adjacent polynomials which correspond to a change in the weight function. A new structure of the singular blocks is given. These results have a direct application to Lanczos methods, theG and -algorithms. 相似文献
4.
E.X.L. de Andrade 《Journal of Mathematical Analysis and Applications》2007,330(1):114-132
We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S3[0,β,b]. Examples are given to illustrate the main contribution in this paper. 相似文献
5.
Orthogonal matrix polynomials, scalar-type Rodrigues’ formulas and Pearson equations 总被引:1,自引:1,他引:0
Some families of orthogonal matrix polynomials satisfying second-order differential equations with coefficients independent of n have recently been introduced (see [Internat. Math. Res. Notices 10 (2004) 461–484]). An important difference with the scalar classical families of Jacobi, Laguerre and Hermite, is that these matrix families do not satisfy scalar type Rodrigues’ formulas of the type (ΦnW)(n)W-1, where Φ is a matrix polynomial of degree not bigger than 2. An example of a modified Rodrigues’ formula, well suited to the matrix case, appears in [Internat. Math. Res. Notices 10 (2004) 482].In this note, we discuss some of the reasons why a second order differential equation with coefficients independent of n does not imply, in the matrix case, a scalar type Rodrigues’ formula and show that scalar type Rodrigues’ formulas are most likely not going to play in the matrix valued case the important role they played in the scalar valued case. We also mention the roles of a scalar-type Pearson equation as well as that of a noncommutative version of it. 相似文献
6.
7.
For discrete multiple orthogonal polynomials such as the multiple Charlier polynomials, the multiple Meixner polynomials, and the multiple Hahn polynomials, we first find a lowering operator and then give a (r+1)th order difference equation by combining the lowering operator with the raising operator. As a corollary, explicit third order difference equations for discrete multiple orthogonal polynomials are given, which was already proved by Van Assche for the multiple Charlier polynomials and the multiple Meixner polynomials. 相似文献
8.
Antonio J. Durán 《Journal of Approximation Theory》2011,163(12):1815-1833
We find structural formulas for a family (Pn)n of matrix polynomials of arbitrary size orthogonal with respect to the weight matrix e−t2eAteA∗t, where A is certain nilpotent matrix. It turns out that this family is a paradigmatic example of the many new phenomena that show the big differences between scalar and matrix orthogonality. Surprisingly, the polynomials Pn, n≥0, form a commuting family. This commuting property is a genuine and miraculous matrix setting because, in general, the coefficients of Pn do not commute with those of Pm, n≠m. 相似文献
9.
Some examples of orthogonal matrix polynomials satisfying odd order differential equations 总被引:2,自引:1,他引:1
It is well known that if a finite order linear differential operator with polynomial coefficients has as eigenfunctions a sequence of orthogonal polynomials with respect to a positive measure (with support in the real line), then its order has to be even. This property no longer holds in the case of orthogonal matrix polynomials. The aim of this paper is to present examples of weight matrices such that the corresponding sequences of matrix orthogonal polynomials are eigenfunctions of certain linear differential operators of odd order. The weight matrices are of the form
W(t)=tαe-teAttBtB*eA*t,