Exceptional Laguerre Polynomials

A simple derivation is presented of the four families of infinitely many shape-invariant Hamiltonians corresponding to the exceptional Laguerre and Jacobi polynomials. The Darboux–Crum transformations are applied to connect the well-known shape-invariant Hamiltonians of the radial oscillator and the Darboux–Pöschl–Teller potential to the shape-invariant potentials of Odake–Sasaki. Dutta and Roy derived the two lowest members of the exceptional Laguerre polynomials by this method. The method is expanded to its full generality and many other ramifications, including the aspects of the generalized Bochner problem and the bispectral property of the exceptional orthogonal polynomials, are discussed.

We survey some recent developments in the theory of orthogonal polynomials defined by differential equations. The key finding is that there exist orthogonal polynomials defined by 2nd order differential equations that fall outside the classical families of Jacobi, Laguerre, and Hermite polynomials. Unlike the classical families, these new examples, called exceptional orthogonal polynomials, feature non-standard polynomial flags; the lowest degree polynomial has degree m > 0. In this paper we review the classification of codimension m = 1 exceptional polynomials, and give a novel, compact proof of the fundamental classification theorem for codimension 1 polynomial flags. As well, we describe the mechanism or rational factorizations of 2nd order operators as the analogue of the Darboux transformation in this context. We finish with the example of higher codimension generalization of Jacobi polynomials and perform the complete analysis of parameter values for which these families have non-singular weights.

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

Many algebraic transformations of the hypergeometric equation σ(x)z"(x) + τ(x)z'(x) + lz(x) = 0, where σ, τ, l are polynomial functions of degrees 2 (at most), 1, 0, respectively, are well known. Some of them involve x = x(t), a polynomial of degree r, in order to recover the Heun equation, extension of the hypergeometric equation by one more singularity. The case r = 2 was investigated by K. Kuiken (see 1979 SIAM J. Math. Anal. 10 (3) 655-657) and extended to r = 3,4, 5 by R. S. Maier (see 2005 J. Differ. Equat. 213 171 - 203). The transformations engendered by the function y(x) = A(x)z(x), also very popular in mathematics and physics, are used to get from the hypergeometric equation, for instance, the Schroedinger equation with appropriate potentials, as well as Heun and confluent Heun equations. This work addresses a generalization of Kimura's approach proposed in 1971, based on differential transformations of the hypergeometric equations involving y(x) = A(x)z(x) + B(x)z'(x). Appropriate choices of A(x) and B(x) permit to retrieve the Heun equations as well as equations for some exceptional polynomials. New relations are obtained for Laguerre and Hermite polynomials.

Convexity of the extreme zeros of Gegenbauer and Laguerre polynomials

Due to practical reasons one is interested inv-stage Runge-Kutta methods whose defining matrix has just one realv-fold eigenvalue. The purpose of this note is to show that methods of this type can be constructed by the method of collocation using the ratio between the zeros of certain Laguerre polynomials as collocation points.

Convexity of the zeros of some orthogonal polynomials and related functions

Liouville's method is applied for investigating the asymptotic behaviour of the Laguerre polynomials. Asymptotic expressions are obtained and a reminder term is estimated. a weight estimate following from these results is obtained.

Let w(x) be the Laguerre weight function, 1 ≤ p < ∞, and Lpw be the space of functions f, p-th power of which is integrable with the weight function w(x) on the non-negative axis. For a given positive integer r, let denote by WrLpw the Sobolev space, which consists of r−1 times continuously differentiable functions f, for which the (r−1)-st derivative is absolutely continuous on an arbitrary segment [a, b] of non-negative axis, and the r-th derivative belongs to the space Lpw. In the case when p = 2 we introduce in the space WrL2w an inner product of Sobolev-type, which makes it a Hilbert space. Further, by lαr,n(x), where n = r, r + 1, ..., we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions lαr,n(x) = xn / n! , where n = 0, 1, r − 1, form a complete and orthonormal system in the space WrL2w. In this paper, the problem of uniform convergence on any segment [0,A] of the Fourier series by this system of polynomials to functions from the Sobolev space WrLpw is considered. Earlier, uniform convergence was established for the case p = 2. In this paper, it is proved that uniform convergence of the Fourier series takes place for p > 2 and does not occur for 1 ≤ p < 2. The proof of convergence is based on the fact that WrLpw ⊂ WrL2w for p > 2. The divergence of the Fourier series by the example of the function ecx using the asymptotic behavior of the Laguerre polynomials is established.

Fisher information of orthogonal hypergeometric polynomials

Consider the model with , where τ is an unknown constant (the boundary of X), Z is a random variable defined on , ε is a symmetric error, and ε and Z are independent. Based on an iid sample of Y, we aim at identifying and estimating the boundary τ when the law of ε is unknown (apart from symmetry) and in particular its variance is unknown. We propose an estimation procedure based on a minimal distance approach and by making use of Laguerre polynomials. Asymptotic results as well as finite sample simulations are shown. The paper also proposes an extension to stochastic frontier analysis, where the model is conditional to observed variables. The model becomes , where Y is a cost, w1 are the observed outputs and w2 represents the observed values of other conditioning variables, so Z is the cost inefficiency. Some simulations illustrate again how the approach works in finite samples, and the proposed procedure is illustrated with data coming from post offices in France.

Asymptotic results on marginal expected shortfalls for dependent risks

We compute the variance asymptotics for the number of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and show that under mild conditions, the asymptotic behavior is the same as in the independent framework. In fact our proof goes beyond this framework and makes explicit the variance asymptotics of various models of random Gaussian processes. Our proof relies on intrinsic properties of the Kac–Rice density in order to give a short and concise proof of variance asymptotics.

The associated Laguerre polynomials LH"(x) and the Hermite polynomials #*(£) occupy a distinctive place in quantum mechanics.A wellestablished relationship exists between Hk(£) and Ln±'b(x); this can be further exploited to obtain generalized Hermite polynomials Hk*(%).The latter, relevant to spatial harmonic oscillators in curvilinear coordinates, are essentially the polynomials obtained by Stroud and Secrest [11] based on transformations of numerical quadratures.The tables presented here are zeros and Gaussian weights of {Ln"(x)\ with a = -0.5(1.0)3.5, re = 4, 8, 16, 32; and those of {#/(!)! with A = 0(1.0)4.0,k = 8, 16, 32, 64.In both cases the results are believed to be correct to one unit in the 25th decimal place.The notations for Lna(x) and Hk(%) employed here are those in Szegö [12] and adopted in Erdélyi et al. [5]; namely a» ^w-("íj.«-»;«+^)-¿(:í:)(x)m/ml, lk/2] (1.2) #*({) = fc!E (-l)m(2£)*-27m!(fc -2m)! \Lna{x)} thus defined forms an orthogonal set in [0, » ) with respect to the weight function e~xx"; Hk(£) are similarly orthogonal in ( -», + oo ) with respect to e~£ .Many physicists, notably Condon and Shortley [3], employ (1.3) Lna(x) = re! ( j iPj( -re + a; a + 1; x) = a polynomial of degree n -a, while many others, e.g., Morse and Feshbach [8] have (1.4)Ln"(x) = (" + «)! (n + a) iFi{-n; a + 1; x).The mathematical statistics literature often employs the notation [km (1.5) Hk(S) = 2~*/2fc!£ (-l)m(V2H)k-2m/ml (fc -2m)! which would correspond to Hek(i) in Erdélyi et al., and /&*(£) in Davis [4], and the corresponding weighting function is e_i /2.

Stationarity testing for nonlinear time series models which include several smooth trend components with (possibly) unknown parameters is considered. A pseudo-Lagrange multiplier stationarity test is proposed and its asymptotic behaviour is derived. The limiting null distribution generally depends on the unknown parameters of the model. A bootstrap approach permits this problem to be circumvented and consistency of the bootstrapped test is obtained. The theoretical analysis is complemented with a simulation study which allows us to check the performance of the test in finite samples. The article ends with an empirical application.