# PDF superior Asymptotic properties of generalized Laguerre orthogonal polynomials

### Asymptotic properties of generalized Laguerre orthogonal polynomials

L (α) n (x) = 2 − 1 π − 1/2 e x/2 ( − x) − α/2 − 1/4 n α/2 − 1/4 e 2( − nx) 1 / 2 1 + O n − 1/2 . (4) This relation holds for x in the complex plane cut along the positive real semiaxis; both ( − x) − α/2 − 1/4 and ( − x) 1/2 must be taken real and positive if x < 0 . The bound of the remainder holds uniformly in every closed domain which does not overlap the positive real semiaxis.

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### The limit relations between generalized orthogonal polynomials.(revised version October 1996)

Firstly, we will consider the case when we add a point mass at x = 0. This case corresponds to the Laguerre, Charlier, Meixner and Kravchuk polynomials. Later on, we will consider the Jacobi and Hahn polynomials which involve two point masses at the ends of the interval of orthogonality. The reason of such a choice of the point in which we will add our posi- tive mass will be clear from formulas (39) and (41) from below, because in such formulas appears the value of the kernel polynomials K n (x;y) and they have a very simple analyti-
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### On the modification of classical orthogonal polynomials: The symmetric case.

In Section 2 we include all the properties of the Hermite and Gegenbauer polynomials which will need. In Section 3 we study the generalized Hermite polynomials and Section 4 is devoted to the Gegenbauer case. In particular, we obtain their expression in terms of the classical polynomials, the hypergeometric representations, the ratio asymptotics, the second order dierential equation and the three-term recurrerence relation that such generalized polynomials satisfy.

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### On characterizations of classical polynomials

It can be shown (see e.g., [3,8,25]) that the only families satisfying the above deﬁnition are the Hermite, Laguerre, Jacobi, and Bessel polynomials. Nevertheless there are other properties characterizing such families and that can be used to deﬁne the classical OPS. The oldest one is the so called Hahn characterization—unless this was ﬁrstly observed and proved for the Jacobi, Laguerre, and Hermite polynomials by Sonin in 1887. In [12], Hahn proved the following, Theorem 1.2 (Sonin–Hahn [12,19]). A given sequence of orthogonal polynomials (P n ) n , is a classical sequence if
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### Distribution of zeros of discrete and continuous polynomials from their recurrence relation.

The hypergeometric polynomials in a continous or a discrete variable, whose canonical forms are the so-called classical orthogonal polynomial systems, are objects which naturally appear in a broad range of physical and mathematical elds from quantum mechanics, the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of Sturm-Liouville dierential and dierence equations. Often, they are encoun- tered in the form of a three term recurrence relation (TTRR) which connects a polynomial of a given order with the polynomial of the contiguous orders. This relation can be directly found, in particular, by use of Lanczos-type methods, tight-binding models or the appli- cation of the conventional discretisation procedures to a given dierential operator. Here the distribution of zeros and its asymptotic limit, characterized by means of its moments around the origin, are found for the continuous classical (Hermite, Laguerre, Jacobi, Bessel) polynomials and for the discrete classical (Charlier, Meixner, Kravchuk, Hahn) polynomials by means of a general procedure which (i) only requires the three-term recurrence relation and (ii) avoids the often high-brow subleties of the potential theoretic considerations used in some recent approaches. The moments are given in an explicit manner which, at times, allows us to recognize the analytical form of the corresponding distribution.
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### On the properties for modifications of classical orthogonal polynomials of discrete variables. (revised version October 1996)

equation. Now we generalize this result for the Kravchuk and Charlier polynomials and continue the algebraic approach presented by Godoy, Marcellan, Salto, Zarzo ( see [7] ) in the framework of a more general theory based in the addition of a delta Dirac measure to a discrete semiclassical linear functional. We analyze the relation between tridiagonal matrices of the perturbed or generalized P An (x) and classical P n (x) polynomials, as well as

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### WKB Approximations and Krall-type orthogonal polynomials.

addition of one or more delta Dirac functions. Some examples studied by dierent authors are considered from an unique point of view. Also some properties of the Krall polynomials are studied. The three-term recurrence relation is calculated explicitly, as well as some asymptotic formulas. With special emphasis will be considered the second order dierential equations that such polynomials satisfy which allows us to obtain the central moments and the WKB approximation of the distribution of zeros. Some examples coming from quadratic transformation polynomial mappings and tridiagonal periodic matrices are also studied.
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### A generalization of the classical Laguerre polynomials: Asymptotic Properties and Zeros.

In section 3 our main aim is concentrated in the location of the zeros of these new orthog- onal polynomials. We deduce that, for n large enough, they are real and simple, n 1 of them are positive and the other one is negative and is attracted by the end of the support with order O ( n

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### Some Extension of the Bessel Type Orthogonal Polynomials.

The structure of the paper is as follows. In Section 2 we list some of the main properties of the classical Bessel polynomials which will be used later on. In Section 3 we dene the generalized polynomials and nd some of their properties. In Section 4 we obtain the representation of the generalized Bessel polynomials in terms of the hypergeometric functions. In Section 5 we obtain an asymptotic formula for these polynomials and in Section 6 we establish their quasi-orthogonality. Finally, in Sections 7 and 8 we obtain the three-term recurrence relation that such polynomials satisfy as well as the corresponding Jacobi matrices.
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### On the Krall-type polynomials

This kind of perturbations can be seen as the simplest (lower order) discrete-continuous version of the “bispectral” property. The continuous “bispectral problem” consists of describing all Schr¨odinger type differential operators of second order such that their eigenfunctions should satisfy a dif- ferential equation (of arbitrary finite order) in the spectral variable. The basic tools are based in the Darboux factorization method [13]. In the discrete- continuous case, this factorization is related to the LU and UL factorization of the Jacobi matrix associated with the sequence of orthogonal polynomials which are the eigenfunctions of a second order linear differential operator of hypergeometric type. Thus the so-called Krall polynomials appear in a natural way [15, 16, 17, 18, 19].
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### Procesos de difusión en una dimensión y polinomios ortogonales

In this work, we will study the so-called diffusion processes in one dimension. The diffusion processes allow us to model some natural phenomena, such as random motion of particles. A diffusion is a random process that has two important features: continuous paths and the Markov property. This last property is characterized by the loss of memory, which means that one can make estimates of the future of the process based solely on its present state just as one could know the full history of the process. In other words, by conditioning on the present state of the system, its future and past are independent.
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### Matrix spherical functions and orthogonal polynomials: an instructive example

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.
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### 8542 pdf

Por lo dem´ as, se han publicado generalizaciones de algunos de los resultados que aparecen aqu´ı. Por ejemplo para el caso del teorema 2.2, en el art´ıculo ”F´ ormulas de cuadratura para intervalos no acotados” publicado en 1982 por el Dr. Ill´ an, se trabaj´ o sobre intervalos no acotados; para el teorema 3.2, en 1996, en el art´ıculo ”The R F -convergence and Theorems of Banach Steinhaus type”, se hizo la generalizaci´ on

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### A q-extension of the generalized Hermite polynomials with the continuous orthogonality property on R

In this paper we study in detail a q-extension of the generalized Hermite polynomials of Szeg˝ o. A continuous orthogonality property on R with respect to the positive weight function is established, a q-difference equation and a three-term recurrence relation are derived for this family of q-polynomials.

### RECURRENCE RELATIONS FOR CONNECTION COEFFICIENTS BETWEEN Q-ORTHOGONAL POLYNOMIALS OF DISCRETE

[8] R. Koekoek and R. F. Swarttouw The Askey-scheme of hypergeometric orthog- onal polynomials and its q-analogue. Reports of the Faculty of Technical Math- ematics and Informatics No. 94-05. Delft University of Technology. Delft 1994. [9] T.H. Koornwinder: Compact quantum groups and q-special functions. In Rep-

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### Higher order hypergeometric Lauricella function and zero asymptotics of orthogonal polynomials

Our aim here is to express in terms of higher order hypergeometric Lauricella functions the corresponding asymptotic contracted measure of zeros for the sequence {P n (x)}™ =1 to be de[r]

### Updating weighting matrices by cross entropy

Some other authors, on the contrary, propose the construction of W matrices based on some «empirical» evidence about the variables of the model. They are critical of the «exogenous approach», because the spatial lag operator imposed can be very dif- ferent from the real spatial structure underlying in the data. For example, Kooijman (1976) or Boots and Dufornaud (1994) define as one criterion the choice of W that maximizes the Moran statistic. Following a similar idea, Mur and Paelinck (2010) base their specification of W on the so-called complete correlation coefficients. Two papers by Getis and Aldstadt base their specification of W on the values of the G* i lo-
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### Black holes within asymptotic safety

tion gives rise to the RG improved radial function shown in Fig. 12. The func- tion smoothly interpolates between classical ﬂat space and a improvement scheme dependent positive constant f (0) < 1. Notably f (r) remains positive throughout so that the RG-improvement process does not lead to the formation of a horizon. At this stage, it is illustrative to investigate the square of the Riemann tensor, obtained from the full RG improved line element. This quantity is shown in Fig. 13 and interpolates smoothly between vanishing curvature for r 1 and a scalar curvature singularity at r = 0. Remarkably, RG improving ﬂat space including a cosmological constant also gives rise to a singularity. Thus, the singularity observed in the last section also appears in the absence of a black hole. Therefore, it is im- portant to carefully distinguish between the classical black hole singularity and the singularity appearing for the quantum improved black hole, since these two eﬀects may come from a very diﬀerent physics origin.
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### On the “Favard theorem” and its extensions

that the main dierence with the recurrence relation analyzed in Section 2 is that here only two consecutive polynomials are involved and the reciprocal polynomial is needed. On the other hand, the basic parameters which appear in these recurrence relations are the value at zero of the orthogonal polynomial.

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### On the Krall-type discrete polynomials

Special cases of quasi-definite linear functionals are the classical ones (those of Ja- cobi, Laguerre, Hermite, and Bessel). In the last years perturbations of the func- tional via the addition of Dirac delta functions —the so-called Krall-type orthog- onal polynomials— have been extensively studied (see e.g. [6, 7, 16, 20, 21, 22, 26] and references therein), i.e., the linear functional

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