Two new associated Laguerre integral results

New associated Laguerre integral results are obtained by eliminating an unnecessary restriction in [1]. The expression used to obtain these new results is compared to that given in [2] and shown to be different and of greater generality. Finally, some physical moments are discussed that require such integrals.     

Spline regularization of numerical inversion of Mellin transform

A method is described for inverting the Mellin transform which uses an expansion in Laguerre polynomials and converts the Mellin transform to the Laplace transform, then the Laplace transform is converted to the first kind convolution integral equation by a suitable substitution. The integral equation so obtained is an ill-posed problem and we use the spline regularization to solve it. The performance of the method is illustrated by the inversion of the test functions available in the literature [J. Inst. Math. & Appl. 20 (1977), p. 73], [J. Math. Comp. 53 (1989), p. 589], [J. Sci. Stat. Comp. 4 (1983), p. 164]. The effectiveness of the method is shown by results obtained demonstrated by means of tables and diagrams.     

On closed-form solutions of triple series equations involving Laguerre polynomials

We consider some triple series equations involving generalized Laguerre polynomials. These equations are reduced to triple integral equations for Bessel functions. The closed-form solutions of the triple integral equations for Bessel functions are obtained and, finally, we get the closed-form solutions of triple series equations for Laguerre polynomials.     

More on Kleinewillinghöfer Types of Flat Laguerre Planes

Polster and Steinke [Result. Math., 46 (2004), 103–122] determined the possible Kleinewillingh?fer types of flat Laguerre planes. These types reflect transitivity properties of groups of certain central automorphisms. We exclude three more types from the list given there with respect to Laguerre homotheties. This yields a complete determination of all possible single types with respect to Laguerre homotheties that can occur in flat Laguerre planes. Building on results by M?urer and Hartmann to characterize ovoidal or miquelian Laguerre planes we further characterize certain flat Laguerre planes in terms of their Kleinewillingh?fer types. Received: January 16, 2007. Revised: July 26, 2007.     

Laguerre functions and representations of su(1,1)

Spectral analysis of a certain doubly infinite Jacobi operator leads to orthogonality relations for confluent hypergeometric functions, which are called Laguerre functions. This doubly infinite Jacobi operator corresponds to the action of a parabolic element of the Lie algebra su(l, 1). The Clebsch-Gordan coefficients for the tensor product representation of a positive and a negative discrete series representation of su(l,l) are determined for the parabolic bases. They turn out to be multiples of Jacobi functions. From the interpretation of Laguerre polynomials and functions as overlap coefficients, we obtain a product formula for the Laguerre polynomials, given by an integral over Laguerre functions, Jacobi functions and continuous dual Hahn polynomials.     

A spectral method of solving two-dimensional integral equations of convolution type

Two-dimensional integral equations of convolution type can be solved by the spectral method in the Chebyshev-Laguerre polynomial basis. We construct an exact parametric representation of the generalized Laguerre spectrum of the required solution with respect to the known Laguerre spectrum of the right-hand side and the kernel of the equation.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 84–88.     

Generalized and mixed type Gegenbauer polynomials

In this paper, a new generalized form of the Gegenbauer polynomials is introduced by using the integral representation method. Further, the Hermite–Gegenbauer and the Laguerre–Gegenbauer polynomials are introduced by using the operational identities associated with the generalized Hermite and Laguerre polynomials of two variables.     

Solution of a certain multiple integral equation

An explicit solution is derived formally for a certain multiple integral equation involving a multidimensional fractional integral of Riemann-Liouville type. The main inversion theorem proved here provides a generalization of a result due to W. L. Wainwright [in “Fractional Calculus and Its Applications” (B. Ross, Ed.), pp. 298–305, Springer-Verlag, Berlin/Heidelberg/New York, 1975]. A simple illustration of the theorem, involving the classical Laguerre function, is also presented.     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

Numerical approximation of weakly singular integrals on the half line

This paper deals with the numerical approximation of a weakly singular integral transform by means of Laguerre nodes. Error estimates in a weighted uniform norm and some numerical tests are given.     

On the Kleinewillinghöfer types of flat Laguerre planes

Kleinewillinghöfer classified in [7] Laguerre planes with respect to central automorphisms and obtained a multitude of types. For finite Laguerre planes many of these types are known to be empty. In this paper we investigate the Kleinewillinghöfer types of flat Laguerre planes with respect to the full automorphism groups of these planes and completely determine all possible types of flat Laguerre planes with respect to Laguerre translations.     

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth-atmosphere model

A numerical-analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth-Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier-Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth-Atmosphere model for different source locations.     

Perturbed Laguerre unitary ensembles,Hankel determinants,and information theory

This article investigates a key information‐theoretic performance metric in multiple‐antenna wireless communications, the so‐called outage probability. The article is partly a review, with the methodology based mainly on [10], while also presenting some new results. The outage probability may be expressed in terms of a moment generating function, which involves a Hankel determinant generated from a perturbed Laguerre weight. For this Hankel determinant, we present two separate integral representations, both involving solutions to certain non‐linear differential equations. In the second case, this is identified with a particular σ‐form of Painlevé V. As an alternative to the Painlevé V, we show that this second integral representation may also be expressed in terms of a non‐linear second order difference equation. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical solutions of the first boundary-value problem for the telegraph equation on open contours

The first plane initial—boundary-value problem for the telegraph equation is reduced by a Chebyshev—Laguerre temporal integral transform to a sequence of stationary boundary-value problems for elliptic equations. Their solutions are sought in integral form. This leads to a recursive sequence of integral equations of the first kind that are solved by the collocation method with isolation of singularities. The sought function is determined by the inverse transform.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 57–62, 1990.     

Numerical solutions of two-dimensional nonlinear integral analysis

In this paper, the approximate solutions for two different type of two-dimensional nonlinear integral equations: two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method. To do this, these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form. By solving these systems, unknown coefficients are obtained. Also, some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method.     

Approximation of analytic functions by Laguerre functions

We will solve the inhomogeneous Laguerre differential equation and apply this result to prove that if a function can be represented by a power series whose radius of convergence is larger than 1, then the function can be approximated, on the interval [0, 1), by a Laguerre function with an error bound C x for some constant C > 0.     

A New Integral Involving the Product of Bessel Functions and Associated Laguerre Polynomials

A new result for integrals involving the product of Bessel functions and Associated Laguerre polynomials is obtained in terms of the hypergeometric function. Some special cases of the general integral lead to interesting finite and infinite series representations of hypergeometric functions.     

Semi Laguerre Planes

A Dembowski semi-plane is a semi-plane obtained from a projective plane by Dembowski's method [1]. A semi Laguerre plane is an incidence structure J = (P, B₁ ∪ B₂, I) for which: (a) every element of P is incident with one element of B₁, (b) an element of B₁ and an element of B₂ are incident with at most one common element of P, (c) each residual space of J (with respect to B₁) is a Dembowski semi-plane, (d) B₂ ≠ ? and each element of B₂ is incident with at least 4 elements of P. We prove that all semi Laguerre planes are substructures of Laguerre planes or special Laguerre planes (in the sense of Thas, Willems [3], [4]). Therefore, these incidence structures are related to optimal codes ([5], [6]).     

Laguerre projection method for finite Hankel transform of arbitrary order

We propose a modification of the projection method for the problem of inverting finite Hankel transform of arbitrary order. In expanding the solution of an integral equation of the first kind, eigenfunctions corresponding to eigenvalues close to the multiple are replaced with Laguerre functions. These functions are eigenfunctions of Hankel transform on the half-line. Our test calculations demonstrated the effectiveness of the elaborated method.