Monte-Carlo computations often yield numerical answers of limited accuracy, and are therefore employed as a last resort. It has been found, however, that some of the limitations of Monte-Carlo methods can be overcome through a judicious use of orthogonal expansions. When a numerical answer is obtained as the expected value of an estimator, expansion of that estimator in a series of orthogonal f...

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials second kind, authors reconsider generating functions Hermite their squares, find an explicit for higher-order derivatives function polynomials, derive formulas recurrence relations squares.

Let Q : R-R be even, nonnegative and continuous, Q0 be continuous, Q040 in ð0;NÞ; and let Q00 be continuous in ð0;NÞ: Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight W 2 rQðxÞ 1⁄4 jxj 2r expð 2QðxÞÞ: In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials fPnðW 2 rQ; xÞg N n1⁄40 with the generalize...

We revisit the radial oscillator from standpoint of a free realization. By using oscillator, namely, creation/annihilation operators harmonic we construct an operator that maps eigenfunctions to those oscillator. As polynomial part this relation, obtain Hermite polynomials Laguerre polynomials.

The operator that intertwines between the $\mathbb{Z}_2$ - Dunkl and derivative is shown to have a realization in terms of oscillator operators one dimension. This observation rests on fact intertwining maps Hermite polynomials generalized polynomials.

We construct solutions of the paraxial and Helmholtz equations that are polynomials in their spatial variables. These are derived explicitly by using the angular spectrum method and generating functions. Paraxial polynomials have the form of homogeneous Hermite and Laguerre polynomials in Cartesian and cylindrical coordinates, respectively, analogous to heat polynomials for the diffusion equati...

We analyze and identify stationary fields with linear regressions and quadratic conditional variances. We give sufficient conditions to determine one dimensional distributions uniquely as normal, and as certain compactly-supported distributions. Our technique relies on orthogonal polynomials, which under our assumptions turn out to be a version of the so called continuous q-Hermite polynomials.

In this work, we give a unification and generalization of Laguerre and Hermite polynomials for which the orthogonal property is replaced by the d-orthogonality. We state some properties of these new polynomials. 2010 Mathematics Subject Classification: Primary 42C05, 33C45, 33C20.

Strong asymptotic results for relativistic Hermite polynomials H n (z) are established as n, N →∞, thereby supplementing recent results on weak asymptotics for these polynomials. Depending on growth properties of the ratio N/n for the rescaled polynomials H n (cnz) (cn being suitable positive numbers, n, N →∞), formulae of Plancherel-Rotach type are derived on the oscillatory interval, in the c...

We express Wronskian Hermite polynomials in the basis and obtain an explicit formula for coefficients. From this we deduce upper bound modulus of roots case partitions length 2. also derive a general real purely imaginary roots. These bounds are very useful study irreducibility polynomials. Additionally, generalize some our results to larger class