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Recently, more and more actuarial researchers have paid attention to valuing life insurance products. In this paper, we study the valuation of equity-linked annuity contracts with guaranteed minimum death benefits (GMDB). The discounted density function approach is used to value this class of life insurance products. We develop the density function of the time-until-death variable on the Laguerre basis and value the GMDB contracts with various equity-linked death benefit functions, including basic options and lookback options. Compared to the Laplace inversion transform method in the literature, our method exhibits remarkable accuracy and efficiency. We also implement life table data to fit the density function of the time-until-death variable by Laguerre functions and provide numerical results for some death benefit functions.

In this paper, we provide a new method for estimating the Gerber–Shiu function in a pure jump Levy risk model. First, we show that the Gerber–Shiu function can be expressed on the Laguerre basis and the Laguerre coefficients can be easily obtained by solving a linear system. Next, based on a high-frequency observation of the aggregate claims process, we estimate the Laguerre coefficients and this leads to a new estimator of the Gerber–Shiu function. We derive the consistency property of this estimator when the sample size is large. Finally, we do some simulation studies to illustrate the finite sample size performance.

The aim of the paper is to establish certain new integrals involving the generalized Gauss hypergeometric function, generalized confluent hypergeometric function, and the Laguerre Polynomials. On account of the most general nature of the functions involved therein, our main findings are capable of yielding a large number of new, interesting, and useful integrals, expansion formulas involving the hypergeometric function, and the Laguerre Polynomials as their special cases.

In this paper, we consider the numerical approximation of the time fractional diffusion equation with variable coefficients on a semi-infinite spatial domain. A fully discrete scheme based on finite difference method in time and spectral approximation using Laguerre functions in space is proposed. Stability and convergence of the proposed scheme are rigorously established. The scheme is unconditionally stable and convergent with order O ( ? 2 + N ( 1 - m ) / 2 ) , where ? , N , and m are the time-step size, polynomial degree, and regularity in the space variable of the exact solution, respectively. A pseudospectral scheme is also proposed and analysed. Some numerical examples are presented to demonstrate the theoretical results.

We state and prove the q -extension of a result due to Johnston and Jordaan (cf. Johnston and Jordaan (2015)) and make use of this result, the orthogonality of q -Laguerre, little q -Jacobi, q -Meixner and Al-Salam–Carlitz I polynomials as well as contiguous relations satisfied by the polynomials, to establish the quasi-orthogonality of certain 2 ϕ 2 and 3 ϕ 2 polynomials. The location and interlacing properties of the real zeros of these quasi-orthogonal polynomials are studied. Interlacing properties of the zeros of q -Laguerre quasi-orthogonal polynomials L n ( δ ) ( z ; q ) when − 2 δ − 1 with those of L n − 1 ( δ + 1 ) ( z ; q ) and L n ( δ + 1 ) ( z ; q ) are also considered.

We propose an iterative method to solve the nonlinear Thomas-Fermi equation based on Laguerre pseudospectral approximation. Different from the direct use of pseudospectral methods for solving the problem in the literature, we represent the solution of the Thomas-Fermi equation as sum of two parts due to its singularity at the origin. Both parts regard the initial slope as a parameter unknown to be determined. One "singular" part is a power-series expansion. The other "smooth" part satisfies a nonlinear two-point boundary value problem. We use a Laguerre pseudospectral method to discretize and solve it numerically by the Newton iteration. Moreover, an introduced new boundary condition is employed innovatively to construct an outer iteration to determine the initial slope. We show numerically this separation of solution as two parts is necessary for improvement of accuracy by direct Laguerre approximations of the Thomas-Fermi equation. Comparisons of our numerical results with those obtained by other methods show effectiveness and accuracy of our approach.

A new effective method for solving Hermitian differential-algebraic systems with constant coefficients is proposed and justified. It uses an explicit representation of the solution based on the spectral pseudo-inversion and an expansion of the matrix exponential via Laguerre polynomial series. An approximate solution is obtained by truncating the series. A new bound for the approximate solution error is derived. Results of numerical experiments with systems arising in interconnect analysis of VLSI circuits are presented and discussed.

In this paper, we solve a characterization problem in the context of the d -orthogonality. That allows us, on one hand, to provide a q -analog for the d -orthogonal polynomials of Laguerre type introduced by the first author and Douak, and on the other hand, to derive new L q -classical d -orthogonal polynomials generalizing the Little q -Laguerre polynomials. Various properties of the resulting basic hypergeometric polynomials are singled out. For d = 1 , we obtain a characterization theorem involving, as far as we know, new L q -classical orthogonal polynomials, for which we give the recurrence relation and the difference equation.

The purpose of this paper is to introduce and discuss a more general class of multiple Hermite polynomials. In this work, the explicit forms, operational formulas and a recurrence relation are obtained. Furthermore, we derive several families of bilinear, bilateral and mixed multilateral finite series relationships and generating functions for the generalized multiple Hermite polynomials.

We find solutions for the diffusion-wave problem in 1D with n-term time fractional derivatives whose orders belong to the intervals (0,1),(1,2) and (0,2) respectively, using the method of the approximation of the convolution by Laguerre polynomials in the space of tempered distributions. This method transfers the diffusion-wave problem into the corresponding infinite system of linear algebraic equations through the coefficients, which are uniquely solvable under some relations between the coefficients with index zero.The method is applicable for nonlinear problems too.

The direct spreading measures of the Laguerre polynomials, which quantify the distribution of its Rakhmanov probability density along the positive real line in various complementary and qualitatively different ways, are investigated. These measures include the familiar root-mean-square or standard deviation and the information-theoretic lengths of Fisher, Renyi and Shannon types. The Fisher length is explicitly given. The Renyi length of order q (such that 2q is a natural number) is also found in terms of the polynomials parameters by means of two error-free computing approaches; one makes use of the Lauricella functions, which is based on the Srivastava-Niukkanen linearization relation of Laguerre polynomials, and another one which utilizes the multivariate Bell polynomials of Combinatorics. The Shannon length cannot be exactly calculated because of its logarithmic-functional form, but its asymptotics is provided and sharp bounds are obtained by use of an information-theoretic optimization procedure. Finally, all these spreading measures are mutually compared and computationally analyzed; in particular, it is found that the apparent quasi-linear relation between the Shannon length and the standard deviation becomes rigorously linear only asymptotically (i.e. for n>>1)., 15 pages, 5 figures, accepted in Journal of Computational and Applied Mathematics

In this paper the authors study “truncated” quadrature rules based on the zeros of Generalized Laguerre polynomials. Then, they prove the stability and the convergence of the introduced integration rules. Some numerical tests confirm the theoretical results.

In the past several years, there has been considerable progress made on a general left-definite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-definite’ has its origins in differential equations but Littlejohn and Wellman [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum {Hr}r>0 of Hilbert spaces and a continuum of {Ar}r>0 of self-adjoint operators. In this paper, we review the main theoretical results in [L. L. Littlejohn, R. Wellman, A general left-definite theory for certain self-adjoint operators with applications to differential equations, J. Differential Equations, 181 (2) (2002) 280–339]; moreover, we apply these results to several specific examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi.

In this paper we propose, a collocation method for solving the Blasius equation. The Blasius equation is a third-order nonlinear ordinary differential equation. This approach is based on a rational scaled generalized Laguerre function collocation method. We also present the comparison of this work with some well-known results and show that the present solution is accurate.

Denote by x"n"k(@a), k=1,...,n, the zeros of the Laguerre polynomial L"n^(^@a^)(x). We establish monotonicity with respect to the parameter @a of certain functions involving x"n"k(@a). As a consequence we obtain sharp upper bounds for the largest zero of L"n^(^@a^)(x).

In this paper we consider the generalized shift operator generated from the Laguerre hypergroup; by means of this, pseudo-differential operators are investigated and Sobolev-boundedness results are obtained.

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between consecutive zeros in several cases.

In this paper, we develop the mixed spectral method for three-dimensional exterior problems, using spherical harmonic and generalized Laguerre functions. Some basic approximation results are established. The mixed spectral schemes are proposed for two model problems. Their convergences are proved. Numerical results demonstrate the efficiency of this new approach.

It is well-known that the classical orthogonal polynomials of Jacobi, Bessel, Laguerre and Hermite are solutions of a Sturm-Liouville problem of the type @s(x)y"n^''[email protected](x)y"n^'[email protected]"ny"n=0,where @s and @t are polynomials such that [email protected]=

The support of the orthogonality measure of so-called little q-Laguerre polynomials {ln(.; a|q)}n=0∞, 0 < q < 1, 0 < a < q-1, is given by Sq = {1, q, q2,...} ∪ {0}. Based on a method of Mlotkowski and Szwarc we deduce a parameter set which admits nonnegative linearization. Moreover, we use this result to prove that little q-Laguerre polynomials constitute a so-called Faber basis in C(Sq).

In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric typeσ(x)yn″(x)+τ(x)yn′(x)-λnyn(x)=0.In this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition.

A new family of generalized Laguerre polynomials is introduced. Various orthogonal projections are investigated. Some approximation results are established. As an example of their important applications, the mixed spherical harmonic-generalized Laguerre approximation is developed. A mixed spectral scheme is proposed for a three-dimensional model problem. Its convergence is proved. Numerical results demonstrate the high accuracy of this new spectral method.

In this paper we prove the existence and uniqueness of the Gaussian interval quadrature formula with respect to the generalized Laguerre weight function. An algorithm for numerical construction has also investigated and some suitable solutions are proposed. A few numerical examples are included.

The authors present a general method of operational nature with a view to investigating the orthogonality properties of several different families of the Hermite and related polynomials. In particular, the classical Hermite polynomials and some of their higher-order and multi-index generalizations are considered here.

Fisher's information and Shannon's entropy are two complementary information measures of a probability distribution. Here, the probability distributions which characterize the quantum-mechanical states of a hydrogenic system are analyzed by means of these two quantities. These distributions are described in terms of Laguerre polynomials and spherical harmonics, whose characteristics are controlled by the three integer quantum numbers of the corresponding states. We have found the explicit expression for the Fisher information, and a lower bound for the Shannon entropy with the help of an isoperimetric inequality.

A suite of Matlab programs has been developed as part of the book “Orthogonal Polynomials: Computation and Approximation” Oxford University Press, Oxford, 2004, by Gautschi. The package contains routines for generating orthogonal polynomials as well as routines dealing with applications. In this paper, a brief review of the first part of the package is given, dealing with procedures for generating the three-term recurrence relation for orthogonal polynomials and more general recurrence relations for Sobolev orthogonal polynomials. Moment-based methods and discretization methods, and their implementation in Matlab, are among the principal topics discussed.

For k ∈ N, we consider the analysis of the classical Laguerre differential expression l -k [y](x)=1/x-ke-x(-(x-k+1e-xy'(x))'rx-ke-xy(x)) (x∈(0,∞)), where r≥ 0 is fixed, in several nonisomorphic Hilbert and Hilbert-Sobolev spaces. In one of these spaces, specifically the Hilbert space L2((0, ∞); x-ke-x), it is well known that the Glazman-Krein-Naimark theory produces a self-adjoint operator A -k , generated by l -k [ċ], that is bounded below by rl, where I is the identity operator on L2((0, ∞); x-ke-x). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {H s,-k =(V s,-k (ċ,ċ) s,-k )} s>0 and left-definite self-adjoint operators {B s,-k } s >0 associated with the pair L2((0, ∞); x-ke-x),A -k ). For A -k and each of the operators B s,-k , it is the case that the tail-end sequence {L n -k}∞ n=k of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces. In 1995, Kwon and Littlejohn introduced a Hilbert-Sobolev space W k [0,∞) in which the entire sequence of Laguerre polynomials is orthonormal. In this paper, we construct a self-adjoint operator in this space, generated by the second-order Laguerre differential expression l -k [ċ], having {L n -k}∞ n=0 as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of W k [0, ∞) with the kth left-definite vector space V k,-k .

Asymptotic expansion for the Laguerre polynomials and for their associated functions is extended to the case of a weight function which is the product of the Laguerre weight function by a polynomial, nonnegative on the interval [0,∞[.

Using an alternative definition of usual Hermite polynomials, two problems in the theory of general Hermite and Laguerre 2D polynomials can be separated with advantage for the further treatment: the introduction of a general 2D matrix in the linear transformation of powers of the components of a 2D vector and the generation of Hermite (or Laguerre) polynomials by applying an integral operator to these powers. The Jacobi polynomials appear in the finite-dimensional irreducible representations of the two-dimensional general linear group GL(2,C).

We consider two partial differential operators D1 and D2 having eigenfunctions defined with Laguerre functions. We give the harmonic analysis associated with D1 and D2. We define and study the Wigner transform associated with D1 and D2, and we prove for this transform an inversion formula. Next we consider classes of symbols which allows us to define the Weyl transform associated with D1 and D2. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we study criterions in term of symbols for the boundedness and compactness of this Weyl transform.

By using Ramanujan's q-extension of the Euler integral representation for the gamma function, we derive the Mellin integral transforms for the families of the discrete q-Hermite II, the Al-Salam–Carlitz II, the big q-Laguerre, the big q-Legendre, the big q-Jacobi and the q-Hahn polynomials.

11 pages, no figures.-- MSC2000 codes: 81V55, 33C45, 81Q10.-- Issue title: "Proceedings of the 6th International Symposium on Orthogonal Polynomials, Special Functions and their Applications" (OPSFA-VI, Rome, Italy, 18-22 June 2001). MR#: MR1985711 (2004f:33026) Zbl#: Zbl 1017.81048 The expansion of a large class of functions in series of linearly varying Laguerre polynomials, i.e., Laguerre polynomials whose parameters are linear functions of the degree, is found by means of the hypergeometric functions approach. This expansion formula is then used to obtain the Brown–Carlitz generating function (which gives a characterization of the exponential function) and the connection formula for these polynomials. Finally, these results are employed to connect the bound states of the quantum–mechanical potentials of Morse and Pöschl–Teller, which are frequently used to describe molecular systems. This work has been partially supported by the Junta de Andalucía, under the research Grants FQM0207 (J.S.D. and J.S.R.) and FQM0229 (P.L.A.); the Spanish MCyT projects BFM2001-3878-C02 (J.S.R., J.S.D. and P.L.A.) and BFM2000-0206-C04-01 (J.S.R.); and the European Union project INTAS 2000-272 (J.S.D. and J.S.R.). Publicado

Some monotonicity results for the function f(α)xn,k(α), where xn,k(α) is the kth zero of generalized Laguerre polynomial Ln(α)(x) and f(α) is a suitable function of α, are derived by means of the classical method of Stiltjes.

A new method of the fourth order for the simultaneous inclusion of simple polynomial zeros is proposed. The presented method is based on the fixed point relation of Laguerre's type and realized in circular complex interval arithmetic. The established initial conditions guarantee the convergence of the considered method and the fourth order of convergence. These conditions are computationally verifiable, which is of practical importance. The presented iteration formula is suitable for the construction of modified interval methods with improved convergence rate with negligible additional operations. Convergence analysis and numerical results are given.

Some of the work on the construction of inequalities and asymptotic approximations for the zeros λn,k(α), k = 1,2 .... ,n, of the Laguerre polynomial Lnα(x) as v = 4n + 2α + 2 → ∞, is reviewed and discussed. The cases when one or both parameters n and α unrestrictedly diverge are considered. Two new uniform asymptotic representations are presented: the first involves the positive zeros of the Bessel function Jα(x), and the second is in terms of the zeros of the Airy function Ai(x). They hold for k= 1,2 .... , [qn] and for k = [pn], [pn] + 1 ..... n, respectively, where p and q are fixed numbers in the interval (0, 1 ). Numerical results and comparisons are provided which favorably justify the consideration of the new approximations formulas.

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.

Let {Snλ) denote the monic orthogonal polynomial sequence with respect to the Sobolev inner product (f,g)s = ∫-∞∞ fgdψ0 + λ ∫-∞∞ f'g'dψ1, where {dψ0, dψ1} is a so-called coherent pair and λ < 0. Then Snλ has n different, real zeros. The position of these zeros with respect to the zeros of other orthogonal polynomials (in particular Laguerre and Jacobi polynomials) is investigated. Coherent pairs are found where the zeros of Sn-1λ separte the zeros of Snλ.

The classical orthogonal polynomials are given as the polynomial solutions Pn(x) of the differential equationσ(x)y″(x)+τ(x)y′(x)+λny(x)=0,where σ(x) turns out to be a polynomial of at most second degree and τ(x) is a polynomial of first degree. In a similar way, the classical discrete orthogonal polynomials are the polynomial solutions of the difference equationσ(x)Δ∇y(x)+τ(x)Δy(x)+λny(x)=0,where Δy(x)=y(x+1)−y(x) and ∇y(x)=y(x)−y(x−1) denote the forward and backward difference operators, respectively. Finally, the classical q-orthogonal polynomials of the Hahn tableau are the polynomial solutions of the q-difference equationσ(x)DqD1/qy(x)+τ(x)Dqy(x)+λq,ny(x)=0,whereDqf(x)=f(qx)−f(x)(q−1)x,q≠1,denotes the q-difference operator. We show by a purely algebraic deduction — without using the orthogonality of the families considered — that a structure formula of the typeσ(x)D1/qPn(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N≔{1,2,3,…})is valid. Moreover, our approach does not only prove this assertion, but generates the form of this structure formula. A similar argument applies to the discrete and continuous cases and yieldsσ(x)∇Pn(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N)andσ(x)Pn′(x)=αnPn+1(x)+βnPn(x)+γnPn−1(x)(n∈N).Whereas the latter formulas are well-known, their previous deduction used the orthogonality property. Hence our approach is also of interest in these cases.

In this paper we continue the study of the q -classical (discrete) polynomials (in the Hahn's sense) started in Medem et al. (this issue, Comput. Appl. Math. 135 (2001) 157–196). Here we will compare our scheme with the well known q -Askey scheme and the Nikiforov–Uvarov tableau. Also, new families of q -polynomials are introduced.

In this note integrals of the type I ij (t;k 1 ,k 2 ,k 3 )= ∫ 0 t e k 3 τ ρ i (τ;k 1 )ρ j (t−τ;k 2 ) d τ, i,j=0,1, with ρ 0 (t;k)=I 0 (2 kt ), ρ 1 (t;k)= k t I 1 (2 kt ), will be evaluated. It is possible to expand Iij in a threefold power expansion in k1t, k2t, k3t, see (25). Moreover, it is also possible to expand these integrals into a single sum with Bessel functions In and Laguerre polynomials Ln, see (45) and (46).

By using some preceding results of Buendia et al., in: Alfaro et al. (Eds.), Orthogonal Polynomials and their Applications, Lecture Notes in Mathematics, Vol. 1329, Springer, Berlin, 1986, pp. 222–235, Ricci, J. Math. Phys. 34 (1993) 4884–4891, Natalini, Calcolo 31 (1994) 127–144, and differential equations of associated (see Belmedhi, Ronveaux, Rend. Mat. 11 (1991) 313–326, Zarzo et al., J. Comput. Appl. Math. 49 (1993) 349–359) and co-recursive (see Ronveaux, Marcellan, J. Comput. Appl. Math. 25 (1989) 105–109, Ronveaux et al., J. Comput. Appl. 59 (1995) 295–328) orthogonal polynomials, we obtain numerical results for Newton sum rules of associated and co-recursive Laguerre, Hermite and Jacobi polynomials.

In this paper we consider polynomials, orthogonal with respect to an inner product which consists of the classical Laguerre inner product combined with two linear perturbations of Sobolev type at x=0. We derive linear differential operators, of a specific form and usually of infinite order, having these polynomials as eigenfunctions. In the case, α is a nonnegative integer one of the operators is of finite order.

Many authors have dealt with problems related to the symmetrization of sequences of orthogonal polynomials on a real line or on a unit circle. Particular aspects are treated for quadratic or cubic decompositions. In this paper, we present a technique that unifies the treatment of these topics for an arbitrary positive order of decomposition n by considering a more general definition of orthogonality. This technique is based on the decomposition with respect to the cyclic group of order n. The main theorem is applied to the orthogonality on (n−1)-symmetric curves. Some particular cases are singled out. These results may be useful in studying Hermite–Pade approximations, vector continued fractions and dynamical systems.

We study the asymptotic zero distribution of Laguerre L (� n) n and generalized Bessel B (� n) n polynomials with the parameter � n varying in such a way that the limit ofn=n exists.Our approach is based on a non-hermitian orthogonality satis8ed by these sequences of polynomials.In the cases that remain open we formulate the corresponding conjectures. c � 2001 Elsevier Science B.V. All rights reserved.

We explain how an inner product derived from a perturbation of a weight function by the addition of a delta distribution is used in the orthogonalization procedure of a sequence of martingales related to a Lévy process. The orthogonalization is done by isometry. The resulting set of pairwise strongly orthogonal martingales involved are used as integrators in the so-called (extended) chaotic representation property. As example, we analyse a Lévy process which is a combination of Brownian motion and the Gamma process and encounter the Laguerre-type polynomials introduced by Littlejohn.

It is shown that an appropriate combination of methods, relevant to generalized operational calculus and to special functions, can be a very useful tool to treat a large body of problems both in physics and mathematics. We discuss operational methods associated with multivariable Hermite, Laguerre, Legendre, and other polynomials to derive a wealth of identities useful in quantum mechanics, electromagnetism, optics, etc., or to derive new identities between special functions as, e.g., Mehler- or mixed-type generating functions.

We use operational identities to introduce multivariable Laguerre polynomials. We explore the wealth of differential equations they satisfy. We analyze their properties and the link with Legendre-type polynomials.

The positivity of a bilinear functional a a(f,g)= ∑ N m=0 λ m C (m) (f m ,g m ) is studied as a function of coefficients λm. The concerned cases are those of Laguerre, Gegenbauer and Jacobi for c(m) = c(0), m = 1,…,N. The domain λmm=1N where a is positive definite, is given. As a consequence, when N = 1, the corresponding Markov-Bernstein inequalities are given.

In this paper we shall make some perturbations in a Laguerre–Hahn linear functional, such as the addition of a Dirac Delta or the left multiplication by a polynomial. We shall study that these transformations carried out on Laguerre–Hahn linear functionals originate new Laguerre–Hahn linear functionals. We shall also analyze the class of the resulting functional.

Limit relations between classical discrete (Charlier. Meixner, Kravchuk, Hahn) to classical continuous (Jacobi, Laguerre, Hcrmite) orthogonal polynomials (to be called transverse limits) are nicely presented in the Askey tableau and can be described by relations of type lim., -0 P n (x(s.ω))=Q n (s). where P n (s) and Q n (s) stand for the discrete and continuous family, respectively. Deeper information on these limiting processes can be obtained by expanding the discrete polynomial family as P n (x(s,ω))= Σ y λ ω k R k (s:n). In this paper a method for computing the coefficients R k (s,n) is designed the main tool being the consideration of some closely related connection and inversion problems.