André Ronveaux

André Ronveaux
Université Catholique de Louvain - UCLouvain | UCLouvain · Institute of Research in Mathematics and Physics

Ing.(E-M) ,M.Sc.(Math.),Ph.D (Phys.)

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134
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Publications

Publications (134)
Article
In this paper we present a recurrent procedure to solve an inversion problem for monic bivariate Krawtchouk polynomials written in vector column form, giving its solution explicitly. As a by-product, a general connection problem between two vector column of monic bivariate Krawtchouk families is also explicitly solved. Moreover, in the non monic ca...
Article
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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 sing...
Article
In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second--order linear partial differential equations, which are admissible potentially self--adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms...
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This paper addresses a general method of polynomial transformation of hypergeometric equations. Examples of some classical special equations of mathematical physics are generated. Heun's equation and exceptional Jacobi polynomials are also treated.
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The eigenmodes of surface plasmon oscillation for an arbitrary number of aligned spheres or spherical cavities are given by a power series of the radius–distance ratio. The Van der Waals energy is calculated in first approximation for an arbitrary chain, and in second approximation for an array of two holes and two spheres.The error due to the negl...
Article
The eigenmode of surface plasmon are obtained for different geometries. The results are valid for voids and for particles and give the eigensurface mode of spheroids, paraboloids, cones, and multi-edges. Limiting cases, obtained by asymptotic development, or particular cases useful in some physical situations are studied in more detail: spheroïdal...
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The equations giving the proper modes of oscillation of the surface plasmons around a cavity in the interior of a semi-infinite metal are established. These modes are calculated to the lowest order for any l, m, and the dominant term in the Van der Waals interaction between the hole and the surface can be obtained from them. The method, which uses...
Article
This paper addresses new results on the factorization of the general Heun’s operator, extending the investigations performed in previous works [the second author, Appl. Math. Comput. 141, No. 1, 177–184 (2003; Zbl 1035.34100), and the authors and K. Sodoga, ibid. 189, No. 1, 816–820 (2007; Zbl 1126.34307)]. Both generalized k-Lamé and k-Heun’s seco...
Article
This paper addresses new results on the factorization of the general Heun's operator, extending the investigations performed in previous works [Applied Mathematics and Computation 141 (2003), 177 - 184 and 189 (2007), 816 - 820]. Both generalized k-Lamé and k-Heun's second order differential equations are considered.
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It is usually claimed that the Laguerre polynomials were popularized by Schrödinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solut...
Article
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This paper addresses new results on the factorization of the general Heun's operator, extending the investigations performed in previous works [{\it Applied Mathematics and Computation} {\bf 141} (2003), 177 - 184 and {\bf 189} (2007), 816 - 820]. Both generalized Heun and generalized Lam\'e equations are considered.
Article
We discuss how the derivative of solutions of some Heun’s differential equations can be given, in some particular cases, from the solution of another Heun’s equation. This work uses some algebraic links between Heun and hypergeometric equations.
Article
Let c n,k (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P n (x). The centroid of these zeroes: controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s n , like bounds, inequalities, parameters dependence, ..., from the links between s n , the coefficients of the expansion of P...
Article
This paper stands for a logic continuation of the previous work [A. Ronveaux, Factorization of the Heun’s differential operator, Appl. Math. Comput. 141 (2003) 177–184]. We propose a factorization of the confluent, double confluent and biconfluent Heun’s differential equations and investigate their solutions.
Chapter
We present a survey on the properties of the Bezout polynomials $A(x)$ and $B(x)$ solving the Bezout's problem $A(x)P_n(x)+B(x)P_n'(x)=1$, when $P_n(x)$ belongs to an orthogonal polynomial family. We extend results given by P. Humbert for Legendre polynomials on the several recurrences involving the four families $P_n(x), P_n'(x), A(x)$ y $B(x)$ an...
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The Kustaanheimo-Stiefel transformation is briefly described in various frameworks. This transformation is used to convert the R 3 harmonics into R 4 harmonics. Then, the Schrödinger equation for an hydrogen-like atom is transformed into the set of a coupled pair of Schrödinger equations for two R 2 isotropic harmonic oscillators and a coupled pair...
Article
In this paper, the Bezout's identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type. Differential equations, relation with the starting family as well as recurrence relations and explicit representations are given for the Bezout's pair. Extensions to classical o...
Article
In this paper, we give Laguerre-Freud equations for the recurrence coefficients of discrete semi-classical orthogonal polynomials of class two, when the polynomials in the Pearson equation are of the same degree. The case of generalized Charlier polynomials is also presented.
Article
This paper provides with a generalization of the work by Wimp and Kiesel [Non-linear recurrence relations and some derived orthogonal polynomials, Ann. Numer. Math. 2 (1995) 169–180] who generated some new orthogonal polynomials from Chebyshev polynomials of second kind. We consider a class of polynomials P˜n(x) defined by: P˜n(x)=(anx+bn)Pn-1(x)+(...
Article
The number of zeros in (-1,1) of the Jacobi function of second kind , α,β>-1, i.e. the second solution of the differential equation(1-x2)y″(x)+(β-α-(α+β+2)x)y′(x)+n(n+α+β+1)y(x)=0,is determined for every n∈N and for all values of the parameters α>-1 and β>-1. It turns out that this number depends essentially on α and β as well as on the specific no...
Article
A Rodrigues type representation for the second kind solution of a second-order q-difference equation of hypergeometric type is given. This representation contains some q-extensions of integrals related with relevant special functions. For these integrals, a general recurrence relation, which only involves the coefficients of the q-difference equati...
Article
We consider a class of polynomials h ˜ n (x;q) defined by h ˜ n (x;q)=(a n x+b n )h n-1 (x;q)+(1-a n )h n (x;q),n=0,1,2,⋯,a 0 ≠1, where h n (x;q) are monic q-discrete Hermite orthogonal polynomials satisfying the following three-term recurrence relation: h n+1 (x;q)=xh n (x;q)+q n-1 (q n -1)h n-1 (x;q), n≥1, h 1 (x;q)=x, h 0 (x;q)=1. We derive expl...
Article
Explicit formulae for the coefficients of expansion of q-Bernstein polynomials in terms of little q-Jacobi polynomials and vice versa are given as evaluations of q-Hahn and dual q-Hahn polynomials, respectively. These results are used to derive a procedure for obtaining N points on a q-Bézier curve of degree n in ℝ, with a cost of O(n + Nnd) operat...
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We derive and factorize the fourth-order q-di#erence equations satisfied by orthogonal polynomials obtained from some perturbations of the recurrence coe#cients of q-classical orthogonal polynomials. These perturbations include the rth associated, the anti-associated, the general co-recursive, co-recursive associated, co-dilated and the general co-...
Article
Following the pressure of Hermite in 1865 and Heine in 1868, the name of Rodrigues was definitively associated to his famous representation given in 1815. In this paper, we precise the contribution of Rodrigues, proving that his formula not only generates the Legendre polynomials, but also special Gegenbauer polynomials and even the corresponding s...
Article
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Classification of polynomial solutions of second-order difference equation of hypergeometric type with real coefficients, orthogonal with respect to a positive symmetric weight function is presented.
Article
We factorize the fourth-order differential equations satisfied by the Laguerre–Hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as: the rth associated (for generic r), the general co-recursive, the general co-recursive associated, the general co-dilated and the general co-modified classical ortho...
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In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechko and certain explicit con- nection coecients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.
Article
A Rodrigues-type representation for the second kind solutions of a second-order diierential equation of hypergeometric type is given. This representation contains some integrals related with relevant special func-tions. For these integrals, a general recurrence relation, which only involves the coeecients of the diierential equation, is also presen...
Article
The differential equation with four regular singularities located at z=0,1,aand∞, called the Heun’s equation (HE), isy″(z)+γz+δz−1+εz−ay′(z)+αβz−qz(z−1)(z−a)y(z)=0with α+β+1=γ+δ+ε, and defines the Heun’s operator H byH[y(z)]={P3(z)D2+P2(z)D+P1(z)}[y(z)]with D≡d/dz and Pi(z) polynomials of degree i.H can be factorized in the formH=[L(z)D+M(z)][L(z)D...
Article
This letter proposes two different approaches in order to solve the connection problems Pn(−ξ)=∑m=0n Cm(n)Pm(ξ), when the family {Pn(x)} belongs to a wide class of polynomials, including classical discrete orthogonal polynomials. The first approach uses the involutory character of the Lah numbers and of other specific connection coefficients. The s...
Article
We use some relations between the rth associated orthogonal polynomials of the Dq-Laguerre–Hahn class to derive the fourth-order q-difference equation satisfied by the co-recursive rth associated orthogonal polynomials of the Dq-Laguerre–Hahn class.When r=1 and for q-semi-classical situations, this q-difference equation factorizes as product of two...
Article
We give explicitly the recurrence coefficients in the three term recurrence relation of some generalized Jacobi polynomials defined by the positive weight ϱ(α,α + p;x,μ) = ‖−μ(1−x2)α(1−x)p on [−1, +1]. The case p = 0 can be found in Chihara's book. The case p = 1 is treated by the first author, and we consider here the cases p = 2,3,4. These recurr...
Article
We propose an approach to develop multivariable polynomials in multiple series of orthogonal polynomials in one variable. The action of a partial differential operator on a series of products of classical orthogonal polynomials is first analyzed and permits the generation of recurrence relations for the expansion coefficients like in the one variab...
Article
For the polynomial families {Pn(x)}n belonging to the Askey scheme or to its q-analogue, the hypergeometric representation provides a natural expansion of the form Pn(x)=∑m=0nDm(n)θm(x), where the expanding basis θm(x) is, in general, a product of Pochhammer symbols or q-shifted factorials. In this paper we solve the corresponding inversion problem...
Article
We derive the fourth-order q-difference equation satisfied by the co-recursive of q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials φ and ψ appearing in the q-Pearson difference equation Dq(φρ)=ψρ defining the weight ρ of the q-classical orthogonal polynomials inside the q-Hahn tableau. Use...
Article
We present a method to develop multivariate polynomials in multiple series of products of univariate classical orthogonal polynomials of a discrete variable. The method is based on the ability to apply difference operators on such series in order to generate recurrence relations for the expansion coefficients like in the one-variable case.
Article
The authors give a detailed general solution of the Dirichlet problem for two exterior touching spheres. From this general result they derive an explicit formula giving electrostatic capacity of two unequal exterior adhering spheres.
Article
Starting from the Dω-Riccati difference equation satisfied by the Stieltjes function of a linear functional, we work out an algorithm which enables us to write the general fourth-order difference equation satisfied by the associated of any integer order of orthogonal polynomials of the Δ -Laguerre–Hahn class. Moreover, in classical situations (Meix...
Article
Polynomial modifications of a classical discrete linear functional are examined in detail, in particular when the new linear functional remains classical. New addition formulas are deduced for Charlier, Meixner and Hahn polynomials from the Christoffei representation and results are also given for a particular generalized Meixner family.
Article
Most of the classical orthogonal polynomials (continuous, discrete and their q-analogues) can be considered as functions of several parameters ci. A systematic study of the variation, infinitesimal and finite, of these polynomials Pn(x,ci) with respect to the parameters ci is proposed. A method to get recurrence relations for connection coefficient...
Article
Full-text available
We propose an algorithm to construct recurrence relations for the coefficients of the Fourier series expansions with respect to the q-classical orthogonal polynomials pk(x;q). Examples dealing with inversion problems, connection between any two sequences of q-classical polynomials, linearization of ϑm(x) pn(x;q), where ϑm(x) is xmor (x;q)m, and the...
Article
Generalized Charlier polynomials are introduced as semi-classical orthogonal polynomials of class 1 with one parameter. Main characteristic data are established from the Laguerre–Freud equations generating the coefficients of the recurrence relation satisfied by the polynomials.
Article
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A procedure is proposed in order to expand w = Q N j=1 Pij(x) = P M k=0 LkPk(x) where Pi(x) belongs to a classical orthogonal polynomial sequence (Jacobi, Bessel, Laguerre and Hermite) (M = P N j=1 ij). We first derive a linear differential equation of order 2N satisfied by w, from which we deduce a recurrence relation in k for the linearization co...
Article
Given a polynomial solution of a differential equation, its m -ary decomposition, i.e. its decomposition as a sum of m polynomials P[ j ](x) =∑kαj,kxλj, kcontaining only exponentsλj, k with λj,k+1−λj,k=m, is considered. A general algorithm is proposed in order to build holonomic equations for the m -ary parts P[ j ](x) starting from the initial one...
Article
If { Pn(x;q)}nis a family of polynomials belonging to the q -Hahn tableau then each polynomial of this family can be written as Pn(x;q) =∑m=0nDm(n)ϑm(x) where ϑm(x) stands for (x;q)mor xm. In this paper we solve the corresponding inversion problem, i.e. we find the explicit expression for the coefficients Im(n) in the expansion ϑn(x) =∑m=0nIm(n)Pm(...
Article
If { Pn(x;q)}nis a family of polynomials belonging to the q -Hahn tableau then each polynomial of this family can be written as Pn(x;q) = ∑m = 0nDm(n)ϑm(x) where ϑm(x) stands for (x;q)mor xm. In this paper we solve the corresponding inversion problem, i.e. we find the explicit expression for the coefficients Im(n) in the expansion ϑn(x) = ∑m = 0nIm...
Article
In this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q-polynomials in the exponential lattice x(s)=c1qs+c3. Some simple examples are worked out in detail.
Article
Full-text available
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (See also 42C05 for general orthogonal polynomials and functions) 65Q05 Difference and functional equations, recurrence relations
Article
We derive the fourth-order q-difference equation satisfied by the first associated of the q-classical orthogonal polynomials. The coefficients of this equation are given in terms of the polynomials σ and τ which appear in the q-Pearson difference equation Dq(σϱ) = τϱ defining the weight ϱ of the q-classical orthogonal polynomials inside the q-Hahn...
Article
Explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials (Charlier, Meixner, Krawtchouck, Hahn, Hahn - Eberlein) into the falling factorial basis are given. The corresponding inversion problems are solved explicitly. This is done by using a general algorithm, recently developed by the authors, which is...
Article
A formulation of the generalized Hurwitz problem is given. The authors shown that the solution of this problem is equivalent to exhibiting a matrix extending Yiu's intercalate matrix (1987) consistently signed. They give examples of explicit solutions for such a generalized Hurwitz problem in low dimensions.
Article
Potentials of the r-(s+2) power type are studied as the superposition of Yukawa potentials. The SL(s,R) group acting on the quadrivector impulsion considered as a quaternion generates a compact self-adjoint operator deduced from the Schrodinger operator by a Fourier-Fock transformation. The operator is approximated by finite rank operators and give...
Article
We prove that if Pi(x) and Pj(x) are two families of semi-classical orthogonal polynomials, all the linearization coefficients Li,j,k occurring in the product of these two families satisfy a linear recurrence relation involving only the k index. This property also extends to the linearization coefficients arising from an arbitrary number of product...
Article
The author gives explicitly the fourth-order differential equation satisfied by the numerator polynomials (associated polynomials) of the classical orthogonal polynomials. The coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau (and their derivatives) defined via the relation ( sigma rho )'...
Article
Exact analytical expressions for the dielectric and Kerr functions in both relaxation and steady-state regimes are explicitly calculated by solving the Fokker-Planck-Kramers (FPK) equation for the rotational Brownian motion of a linear rigid rotor in 3D. The response functions thus obtained generalize and extend all the results recently published o...
Article
We consider nonhomogeneous hypergeometric-type differential, difference and q-difference equations whose nonhomogeneity is a polynomial qn(x). The polynomial solution of these problems is expanded in the basis, and also in a basis , related in a natural way with the homogeneous hypergeometric equation. We give an algorithm building a recurrence rel...
Article
The Laguerre-Freud equations giving the recurrence coefficients βn, γn of orthogonal polynomials with respect to a Dω semi-classical linear form are derived. Dω is the difference operator. The limit when ω → 0 are also investigated recovering known results. Applications to generalized Meixner polynomials of class one are also treated.
Article
Limit relations between classical discrete (Charlier, Meixner, Kravchuk, Hahn) to classical continuous (Jacobi, Laguerre, Hermite) orthogonal polynomials (to be called transverse limits) are nicely presented in the Askey tableau and can be described by relations of type limω→0 Pn(x(s, ω)) = Qn(s), where Pn(s) and Qn(s) stand for the discrete and co...
Article
Expansions of continuous and discrete Bernsein bases on shifted Jacobi and Hahn polynomials, respectively, are explicitly obtained in terms of Hahn-Eberlein orthogonal polynomials. The basic tool is an algorighm, recently developed by the authors, which allows one to solve the connection problem between two families of polynomials recurrently. This...
Article
We derive the fourth-order difference equation satisfied by the first associated of classical orthogonal polynomials of a discrete variable. We give it explicitly for first associated of Hahn polynomials from which can be derived by a limiting process the equation satisfied by first associated of all classical families (continuous and discrete).
Article
Limit relations between classical continuous (Jacobi, Laguerre, Hermite) and discrete (Charlier, Meixner, Kravchuk, Hahn) orthogonal polynomials are well known and can be described by relations of type limλ → ∞Pn(x;λ) = Qn(x). Deeper information on these limiting processes can be obtained from the expansion . In this paper a method for the recursiv...
Article
We present a simple approach in order to compute recursively the connection coefficients between two families of classical (discrete) orthogonal polynomials (Charlier, Meixner, Kravchuk, Hahn), i.e., the coefficients Cm(n) in the expression Pn(x) = ∑m=0nCm(n)Qm(x), where {Pn(x)} and {Qm(x)} belong to the aforementioned class of polynomials. This is...
Article
We derive the fourth-order difference equation satisfied by the associated order r of classical orthogonal polynomials of a discrete variable. The coefficients of this equation are given in terms of the polynomials # and # which appear in the discrete Pearson equation #(##)=## defining the weight #(x) of the classical discrete orthogonal polynomial...
Article
We present a simple approach in order to compute recursively the connection coefficients between two families of classical (continuous) orthogonal polynomials (Hermite, Laguerre, Jacobi and Bessel), i.e. the coefficients Cm(n) in the expression , where {Pn(x)} and {Qm(x)} belong to the aforementioned class of polynomials. This is done by adapting a...
Article
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Let fPn (x)g 1 n=0 and fQm (x)g 1 m=0 be two families of orthogonal polynomials. The linearization problem involves only one family via the relation: P i (x) P j (x) = i+j X k=jiGammajj L ijk P k (x) and the connection problem mixes both families: Pn (x) = n X m=0 Cm (n) Qm (x): In many cases, it is possible to build a recurrence relation involving...
Article
An explicit representation of the associated Meixner polynomials (with a real association parameter γ⩾0) is given in terms of hypergeometric functions. This representation allows to derive the fourth-order difference equation verified by these polynomials. Appropriate limits give the fourth-order difference equation for the associated Charlier poly...
Article
We introduce a system of “classical” polynomials of simultaneous orthogonality, study the differential equation of third order, recurrence relation and precise the ratio asymptotic and zeros distribution of polynomials.
Article
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We discuss the difference equations on a linear lattice for polynomials associated with the classical Hahn, Kravchuk, Meixner, and Charlier polynomials.
Article
Sobolev type orthogonal polynomials have been the object of increasing interest in the last few years. In this paper we introduce a generalization of the usual Sobolev-type inner product and we compare it with the strict diagonal case. Zeros and asymptotic properties of these kinds of polynomial sequence are studied.
Article
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix $r$ new rows and columns, so that the orig...
Article
The nodal structure of the wavefunctions of a large class of quantum‐mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials. It is known that these polynomials (i) may be described by an arbitrary linear combination of two orthogonal polynomials {Pn(x)} and (ii) have real and simple zeros. Here, the...
Article
We describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second- and fourth-order differential equations.
Article
The unique fourth-order differential equation satisfied by the generalized co-recursive of all classical orthogonal polynomials is given for any (but fixed) level of recursivity. Up to now, these differential equations were known only for each classical family separately and also for a specific recursivity level. Moreover, we use this unique fourth...
Article
Stieltjes considered sums of reciprocals of differences of zeros of a solution of a homogeneous second-order linear differential equation. Here we re-examine the derivation of these sums with a view to extending the class of differential equations to which the theory applies and including sums involving the zeros of the derivatives as well as those...
Article
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The purpose of the paper is to give information about zeros of orthogonal polynomials. We give special importance to the method based on differential-difference relation (structure relation) satisfied by semi-classical orthogonal polynomials. We do not discuss other methods (such as those based on recurrence relations, weight functions and differen...
Article
We use real representations of Clifford algebras associated with Hurwitz factorizations in order to generate harmonic morphisms on pseudo-Euclidean spaces and generalized Hadamard matrices.
Article
In this paper we give a new scheme for deriving a non-linear system, satisfied by the three-term recurrence coefficients of semi-classical orthogonal Polynomials, this non-linear system is labelled "Laguerre-Freud′s equations." Here we do not deal with the numerical aspect of the question (stability, asymptotic, ...). Our purpose is to take due adv...
Article
The first associated (numerator polynomials) of all classical orthogonal polynomials satisfy one fourth-order differential equation valid for the four classical families, but for the associated of arbitrary order the differential equations are only known separately. In this work we introduce a program built in Mathematica symbolic language which is...
Article
In this paper the inner product $\langle {f,g} \rangle = \int_I {fg\,d\mu } + Mf(c)g(c) + Nf'(c)g'(c)$ is considered, where $\mu $ is a positive measure on the interval I, $c \in {\bf R}$ and M, $N \geqq 0$. General algebraic properties of the orthogonal polynomials associated with $\langle { \cdot , \cdot } \rangle $ as well as the zeros and their...
Article
The linear form sum of two semiclassical regular linear forms verifies in general a second-order differential equation and we examine some situations where this new form remains semiclassical. The sequences of polynomials orthogonal with respect to this new form, called of second category, have associated polynomials which do not belong to the Lagu...
Article
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The symmetric convolution of two Hermite functions appears as transition amplitude in the multiphotonic bremsstrahlung effect analysed by quantum oscillator models. The authors identify these transition amplitudes in terms of Laguerre polynomials and give new recurrence relations and a differential equation for these physical quantities.
Article
The steady state response of the nonlinear Kerr effect to a small alternating field superimposed on a unidirectional field is analyzed using a second order perturbation theory. We draw inferences from the appearance of simple and double harmonics as the fundamental frequencies of the nonlinear Kerr effect. The study is based on the Smoluchowski equ...
Article
This paper analyzes polynomials orthogonal with respect to the Sobolev inner product φ{symbol} ̃(f,g)= ∫ Rf(x)g(x)ρ{variant}(x)dx+λ-1f(r)(c)g(r)(c)withλ∈R+, c∈R, and ρ{variant}(x)is a weight function. We study this family of orthogonal polynomials, as linked to the polynomials orthogonal with respect to ρ{variant}(x) and we find the recurrence rela...
Article
The distribution of zeros of the semiclassical orthogonal polynomials with weights , wc(x) denoting a classical weight function and π(x) being equal to |x − c| or x2 + c2, is investigated via the Newton sum rules of zeros (i.e, the sums of the rth powers of zeros). Recursion relations satisfied by these sum rules for semiclassical Legendre, Laguerr...
Article
Orthogonal polynomials may be fully characterized by the following recurrence relation: Pn(x) = (x − βn-1)Pn-1(x)-γn-1Pn-2(x), with P0(x)=1, P1(x) = x - β0 and γn ≠ 0. Here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the β and γ parameters of a co-recursive (βk → βk + μ), co-dilated (γk →...
Article
We give a new derivation of the fourth-order differential equation satisfied by the co-modified (orthogonal polynomials) of any semi-classical family of orthogonal polynomials.This procedure generalizes a technique already used in the co-recursive and co-dilated cases. For the classical cases we give explicitly the corresponding differential equati...
Article
The following theorem is proved: “Quasi-orthogonal polynomials corresponding to the semi-classical or Laguerre-Hahn families and the orthogonal polynomials themselves, verify differential equations of same order (reducible to two or four by the Hahn theorem)”.
Article
On generalise, de deux facons differentes, l'equation differentielle d'ordre 2 de Little John-Shore pour des polynomes orthogonaux du type Laguerre. D'abord la mesure de Dirac positive peut etre situee en tout point, puis le poids peut etre tout poids classique modifie par un nombre quelconque de distributions de Dirac