On the classical {$d$}-orthogonal polynomials defined by certain generating functions. I (original) (raw)
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.
Generating Function: Multiple Orthogonal Polynomials
In this paper we present a general methodology to obtain a generating function for some multiple orthogonal polynomials of type I with regular indices. In particular, we obtain an explicit generating functions P x (t) := ∞ n=0 Q 2n (x)t n and I x (t) := ∞ n=0 Q 2n+1 (x)t n with Q n (x) = q n,1 (x) + x q n,2 (x) where q n,1 (x), q n,2 (x) is the r-vector of type I associated with the multiple orthogonal Hermite polynomial with regular index for r = 2. 762 W. Carballosa et al.
SOME BASIC d-ORTHOGONAL POLYNOMIAL SETS
The purpose of this paper is to study the class of polynomial sets which are at the same time d-orthogonal and q-Appell. By a linear change of variable, the resulting set reduces to q-Al-Salam-Carlitz polynomials, for d = 1. Various properties of the obtained polynomials are singled out: a generating function, a recurrence relation of order d + 1. We also explicitly express a d-dimensional functional for which the d-orthogonality holds.
On Finite Classes of Two-Variable Orthogonal Polynomials
Bulletin of the Iranian Mathematical Society, 2019
The purpose of this paper is to introduce several finite sets of orthogonal polynomials in two variables, and investigate some general properties of them such as recurrence relations, generating functions, differential equations, and Rodrigues type representations.
Ond-orthogonal polynomials of Sheffer type
Journal of Difference Equations and Applications, 2018
The polynomial sequences of Sheffer type {P n } n≥0 are defined by the following generating function: ∞ n=0 P n (x) n! t n = A(t)e xC(t). In this work, we are interested, with these sequences when they are also d-orthogonal polynomial sets, that is to say polynomials satisfying one standard (d + 1)-order recurrence relation. We revisit some families in the literature and we state an explicit formula giving the exact number of Sheffer type d-orthogonal sets. We investigate, in details, the d-symmetric case and the particular case d = 3.
The relation of the d-orthogonal polynomials to the Appell polynomials
Journal of Computational and Applied Mathematics, 1996
We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials. A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.