On the classical {$d$}-orthogonal polynomials defined by certain generating functions. I (original) (raw)
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d-Orthogonal Analogs of Classical Orthogonal Polynomials
Symmetry, Integrability and Geometry: Methods and Applications, 2018
Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.
On 2-orthogonal polynomials of Laguerre type
International Journal of Mathematics and Mathematical Sciences, 1999
Let{Pn}n≥0be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0andω1(see Definition 1.1). Now, let{Qn}n≥0be the sequence of polynomials defined byQn:=(n+1)−1P′n+1,n≥0. When{Qn}n≥0is, also, 2-orthogonal,{Pn}n≥0is called classical (in the sense of having the Hahn property). In this case, both{Pn}n≥0and{Qn}n≥0satisfy a third-order recurrence relation (see below). Working on the recurrence coefficients, under certain assumptions and well-chosen parameters, a classical family of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionalsω0andω1and obtain their weight functions which satisfy a second-order differential equation. From all these properties, we show that the resulting polynomials are an ...
© Electronic Publishing House ON 2-ORTHOGONAL POLYNOMIALS OF LAGUERRE TYPE
1998
Abstract. Let {Pn}n≥0 be a sequence of 2-orthogonal monic polynomials relative to linear functionalsω0 andω1 (see Definition 1.1). Now, let {Qn}n≥0 be the sequence of polynomi-als defined by Qn: = (n+1)−1P ′n+1, n ≥ 0. When {Qn}n≥0 is, also, 2-orthogonal, {Pn}n≥0 is called “classical ” (in the sense of having the Hahn property). In this case, both {Pn}n≥0 and {Qn}n≥0 satisfy a third-order recurrence relation (see below). Working on the recur-rence coefficients, under certain assumptions and well-chosen parameters, a classical fam-ily of 2-orthogonal polynomials is presented. Their recurrence coefficients are explicitly determined. A generating function, a third-order differential equation, and a differential-recurrence relation satisfied by these polynomials are obtained. We, also, give integral representations of the two corresponding linear functionals ω0 and ω1 and obtain their weight functions which satisfy a second-order differential equation. From all these prop-erties, we sho...
On the associated orthogonal polynomials
Journal of Computational and Applied Mathematics, 1990
By using the second-order recurrence relation this paper gives some new results on associated orthogonal polynomials without referring to the continued fractions' tool. Some results are very useful for obtaining the second-order differential equation satisfied by the semi-classical orthogonal polynomials (Hendriksen and van Rossum (1985) Maroni (1987)) (cf. Section 3). Also, the main formula derives from Proposition 2.6, by which the fourth-order differential equation, satisfied by some Laguerre-Hahn polynomials (Magnus (1984)), is obtained (cf. Behnehdi and Ronveaux (1989), Dini et al. (1989), Ronveaux et al. (1990)).
On ddd-orthogonal Tchebychev polynomials, II
Methods and Applications of Analysis, 1997
In Part I, we considered the following problem: Find all d-orthogonal polynomial sequences {Pn}n>0 such that P^ ^ = P n , n = 0,1,2,..., where {Pn }n>0 is the associated polynomial sequence of {Pn}n>0-The resulting polynomials are an extension of the classical Tchebychev polynomials of the second kind. A detailed study was made in the particular case d = 2. The purpose of this paper is to make similar investigations by considering the analogous problem: Find all d-orthogonal polynomial sequences {Pn}n>0 such that pW = Q n , n = 0,1,2,..., where Qn(x) := (n + l)~1P4 +1 (a;), n > 0. Here, we show that the resulting polynomials are a natural extension of the classical Tchebychev polynomials of the first kind. The recurrence coefficients and generating function are determined explicitly. As in Part I, a detailed study will be carried out for the case d = 2. A third-order differential equation and a differential-recurrence relation are given. Again, in the particular case, integral representations are obtained of the linear functionals with respect to which the resulting polynomials are 2-orthogonal.
d-Orthogonality via generating functions
Journal of Computational and Applied Mathematics, 2007
In this paper, starting from a suitable generating function of a polynomial set, we show how to decide whether the considered polynomial set is d-orthogonal and, if it is so, how to determine the corresponding d-dimensional functional vector. Then, we apply the obtained results to some known and new d-orthogonal polynomial sets. For the known ones, we give new proofs
2. Review of orthogonal polynomials
De Gruyter eBooks, 2017
Review of orthogonal polynomials 2.1 Introduction Developments and interests in orthogonal polynomials have seen continuous and great progress since their appearance. Orthogonal polynomials are connected with many mathematical, physical, engineering, and computer sciences topics, such as trigonometry, hypergeometric series, special and elliptic functions, continued fractions, interpolation, quantum mechanics, partial differential equations. They are also be found in scattering theory, automatic control, signal analysis, potential theory, approximation theory, and numerical analysis. Orthogonal polynomials are special polynomials that are orthogonal with respect to some special weights allowing them to satisfy some properties that are not generally fulfilled with other polynomials or functions. Such properties have made them wellknown candidates to resolve enormous problems in physics, probability, statistics and other fields. Since their origin in the early 19th century, orthogonal polynomials have formed a somehow classical topic related to Legendre polynomials, Stieltjes' continued fractions, and the work of Gauss, Jacobi, and Christoffel, which has been generalized by Chebyshev, Heine, Szegö, Markov, and others. The most popular orthogonal polynomials are Jacobi, Laguerre, Hermite polynomials, and their special relatives, such as Gegenbauer, Chebyshev, and Legendre polynomials. An extending family has been developed from the work of Wilson, inducing a special set of orthogonal polynomials known by his name, which generalizes the Jacobi class. This new family has given rise to other previously unknown sets of orthogonal polynomials, including Meixner Pollaczek, Hahn, and Askey polynomials. Orthogonal polynomials may also be classified according to the measure applied to define the orthogonality. In this context, we cite the class of discrete orthogonal polynomials that form a special case based on some discrete measure. The most common are Racah polynomials, Hahn polynomials, and their dual class, which in turn include Meixner, Krawtchouk, and Charlier polynomials. Already with the classification of orthogonal polynomials, one can distinguish circular and generally spherical orthogonal polynomials, which consists of some special sets related to measures supported by the circle or the sphere. One well-known class is composed of Rogers-Szegö polynomials on the unit circle and Zernike polynomials, which are related to the unit disk. Orthogonal polynomials, and especially classical ones, can generally be introduced by three principal methods. A first method is based on the Rodrigues formula which consists of introducing orthogonal polynomials as outputs of a derivation.
Journal of Mathematical Analysis and Applications, 2012
Two families of d-orthogonal polynomials related to su(2) are identified and studied. The algebraic setting allows for their full characterization (explicit expressions, recurrence relations, difference equations, generating functions, etc.). In the limit where su(2) contracts to the Heisenberg-Weyl algebra h 1 , the polynomials tend to the standard Meixner polynomials and d-Charlier polynomials, respectively.
d-orthogonality of Littleq-Laguerre type polynomials
Journal of Computational and Applied Mathematics, 2011
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.