Orthogonal Polynomials

A method for discretizing the continuum by using a transformed harmonic oscillator basis has recently been presented 关 Phys. Rev. A 63, 052111 共 2001 兲兴 . In the present paper, we propose a generalization of that formal- ism which does not rely on the harmonic oscillator for the inclusion of the continuum in the study of weakly bound systems. In particular, we construct wave functions that represent the continuum by making use of families of orthogonal polynomials whose weight function is the square of the ground state wave function, expressed in terms of a suitably scaled variable. As an illustration, the formalism is applied to one-dimensional Morse, Po¨schl-Teller, and square well potentials. We show how the method can deal with potentials having several bound states, and for the square well case we present a comparison of the discretized and exact continuum wave functions.

To conclude this paper, let us mention that all formulas connecting the different families of orthogonal polynomials here, as well as the new expressions for the Clebsch–Gordan coefficients and the 6j-symbols, can be obtained by using the Whipple’s transformation or Sear’s transformation for hypergeometric and basic hypergeometric series. Our main aim here, however, is to show that it can be done using the already classical theory of orthogonal polynomials of discrete variables developed in [7, 8] in a completely equivalent way.

addition of a mass point. All the formulas and other properties for the classical Meixner, Kravchuk and Charlier polynomials can be found in a lot of books ( see for instance the excellent monograph Orthogonal Polynomials in Discrete Variables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov [11], Chapter 2.)

[11] L. Golinskii and V. Totik, Orthogonal polynomials: from Jacobi to Simon, in Spec- tral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday, P. Deift, F. Gesztesy, P. Perry, and W. Schlag (eds.), Proceedings of Symposia in Pure Mathematics, 76, Amer. Math. Soc., Providence, RI, 2007, pp. 821-874.

(a small subset of the q-world). Here we need to point out that exits two dierent point of view in the study of the q-polynomials. The rst one, in the framework of the q-basic hypergeometric series [6], [8], [9] and the second, in the framework of the theory of dierence equations developed by Nikiforov et al. [12], [13], [14]. In this work we will use the second one because it gives us the possibility to provide an uniform treatment of several classes of orthogonal polynomials and, probably, it is the best way to nd further applications.

The aim of the present contribution is to obtain an analogue of the Askey tableau for such a kind of generalized polynomials with the description of the continuous generalized orthogonal polynomials as limit case of the discrete generalized orthogonal polynomials. Furthermore, we deduce the explicit second order linear dierential equations for two examples which attracted the interest of the researchers: the Laguerre [13] and the Jacobi [19] case. In Section 2 we present a summary of the more useful properties of classical polynomials both in the discrete and continuous case.

is deduced. Finally, in Section 6 a general algorithm in order to generate the second order linear dierential equations that such polynomials satisfy is given. This result is basic for the development of the Section 8, more precisely for the WKB method, in order to obtain the distribution of their zeros. In Section 7, some asymptotic formulas, useful in the study of the zeros, are presented. Finally, in Section 8 we obtain the speed of convergence of those zeros located outside [-1, 1]. On the other hand, we show some graphics concerning the WKB density as well as the analytic behaviour of the distribution of zeros for Jacobi-Sobolev-type orthogonal polynomials.

The analysis of properties of polynomials orthogonal with respect to a perturbation of a measure via the addition of mass points was introduced by P.Nevai [23]. There the asymptotic properties of the new polynomials have been considered. In particular, he proved the dependence of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures sup- ported in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three-term recurrence relation that the corresponding sequence of orthogonal polynomials satisfy.

Abstract. In the scalar case, it is well known that the zonal spherical func- tions of any compact Riemannian symmetric space of rank one can be ex- pressed in terms of the Jacobi polynomials. The main purpose of this paper is to revisit the matrix valued spherical functions associated to the complex projective plane to exhibit the interplay among these functions, the matrix hypergeometric functions and the matrix orthogonal polynomials. We also obtain very explicit expressions for the entries of the spherical functions in the case of 2 × 2 matrices and exhibit a natural sequence of matrix orthogonal polynomials, beyond the group parameters.

There is a well known equivalence between determining multiple orthogonal polynomials and Hermite-Pad´ e approximants of types I and II, see [9, 10, 11]. To see how to generate the multiple orthogonal polynomials of types I and II, and the associated polynomials of the second kind, see [2, 3, 4, 8, 10, 11].

has obtained three new classes of polynomials orthogonal with respect to measures which are not absolutely continuous with respect to the Lebesgue measure. In fact, his study is related to an extension of the very well known characterization of classical orthogonal poly- nomials by S. Bochner. This kind of measures was not considered in [39]. Moreover, in his paper H. L. Krall obtain that these three new families of orthogonal polynomials satisfy a fourth order dierential equation with polynomial coecients. The corresponding measures are given in table 1. A dierent approach to this subject was presented in [28].

ing orthogonal polynomials with respect to the inner product (2), usually called the discrete Sobolev-type Laguerre which constitutes an instance of a larger class of orthogonal polynomi- als: the discrete Sobolev-type orthogonal polynomials. For more detailed description of this Sobolev-type orthogonal polynomials (including the continuous ones) we refer the readers to the recent reviews [12, 14, 15].

useful to obtain the generalized polynomials orthogonal with respect to the quasi-denite moment functional (see Eq. (21) below). All the formulas and properties for the classical Bessel polynomials can be found in [6] and [12]. In this work we will use monic polynomials, i.e., polynomials with leading coecients equal to 1.

This is a a very important gap, specially when derivatives of Dirac delta functionals are considered, because necessary and sufficient conditions for the existence of the corresponding sequence of orthogonal polynomials are needed. They obtain particular cases of our previous results [2, 3, 4, 5, 7] using a different approach based in the above mentioned Heine formula.

of such properties in terms of the location of the mass points with respect to the support of the measure. Particular emphasis was given to measures supported in [ 1 ; 1] and satisfying some extra conditions in terms of the parameters of the three term recurrence relation which the corresponding sequence of orthogonal polynomials satisfy.

In this work we will present a di<erent approach: It can be considered a pure algebraic approach and constitutes an alternative to the two previous ones, and, in some sense is the continuation of the Hahn’s work [16]. Furthermore, we will prove here that the q-classical polynomials are characterized by several relations, analogue to the ones satis1ed by the classical “continuous” (Jacobi, Bessel, Laguerre, Hermite) and “discrete” (Hahn, Meixner, Kravchuk and Charlier) orthogonal polynomials [1,13,21,22] and references therein. Besides, our point of viewis very di<erent from the previous ones based on the basic hypergeometric series and the di<erence equation, respectively. In fact we start with the distributional equation that the q-moment functionals satisfy and we will prove all the other characterizations using basically the algebraic theory developed by Maroni [23]. So, somehow, this paper is the natural continuation of the study started in [22,13] for the “continuous” and “discrete” orthogonal polynomials, respectively. Another advantage of this approach is the uni1ed and simple treatment of the q-polynomials where all the information is obtained from the coeMcients of the polynomials and of the distributional or Pearson equation (compare it with the method by the American school [20] or the Russian ones [29]).

Abstract. Given a perturbation of a matrix measure supported on the unit circle, we analyze the perturbation obtained on the corresponding matrix mea- sure on the real line, when both measures are related through the Szeg˝ o matrix transformation. Moreover, using the connection formulas for the correspond- ing sequences of matrix orthogonal polynomials, we deduce several properties such as relations between the corresponding norms. We illustrate the obtained results with an example.

These kind of polynomials appear as eigenfunctions of a fourth order linear differential operator with polynomial coefficients that do not depend on the degree of the polynomials. They were firstly considered by Krall in [27] (for a more recent reviews see [8] and [26, chapter XV]). In fact, H. L. Krall discovered that there are only three extra families of orthogonal polynomials apart from the classical polynomials of Hermite, Laguerre and Jacobi that satisfy such a fourth order differential equation which are orthogonal with respect to measures that are not absolutely continuous with respect to the Lebesgue measure. Namely, the Jacobi-type polynomials that are orthogonal with respect to the weight function ρ(x) = (1 − x) α +M δ(x), M > 0, α > −1 supported on [0, 1], the Legendre-type polynomials orthogonal on [−1, 1] with respect to ρ(x) = α/2+δ(x − 1)/2+δ(x + 1)/2, α > 0, and the Laguerre-type polynomials that are orthogonal with respect to ρ(x)e −x + M δ(x), M > 0 on [0, ∞). This result motivated the study of the polynomials orthogonal with respect to the more general weight functions [23, 25] that could contain more instances of orthogonal polynomials being eigenfunctions of higher- order differential equations (see also [26, chapters XVI, XVII]).

We describe the general theory of diffusion processes, which contains as a particular case the solutions of stochastic differential equations. The idea of the theory is to construct explicitly the generator of the Markov process using the so-called scale function and the speed measure. We also explain how the theory of orthogonal polynomials help to study some diffusions. In addition, using the theory of diffusions, we present the Brox model, which is a process in a random environment.

Let us also point out that there are also the so-called discrete (see e.g. [8]) and q-discrete Sobolev type orthogonal polynomials associated with the classical discrete and q-classical functionals [23, 24]. In both cases the corresponding polynomials can be reduced to the Krall-type one (except for the q-case when the mass is added at zero where a more careful study is needed [23, 24]) since the differences ∆f (x) = f (x + 1) − f(x) and D q f(x) = (f (qx) − f(x))/(qx − x).