d-Orthogonality of Hermite type polynomials
Section snippets
Introduction and main result
Many papers deal with characterization problems which consist to find all orthogonal polynomials satisfying a fixed property. Al-Salam [3] gave a large survey on this subject. The Hermite polynomials appeared in various characterization theorems for orthogonal polynomials (see, for instance [2], [4], [18], [19], [26], [31], [36], [37], [41], [42]). In this paper, we will show among others that this polynomial set is one of the solutions of a further characterization problem. Our starting point
Proof of Theorem 1.1
To prove Theorem 1.1, we need the three following lemmas. Lemma 2.1 Let be the sequence given by (1.3) with . Then we have being the falling factorial polynomials given by Proof The relations (2.1), (2.2) can be deduced from the identities: Lemma 2.2 Let be the d-OPS defined by (1.1). Then there
Two characterization theorems
In this section, we use Theorem 1.1 to characterize d-OPSs having hypergeometric representations of the form (1.5), (1.6). To this end, we need the following. Definition 3.1 Let be a set of s polynomials of degree one, . is called a s-separable product set if and only if there exists a polynomial such that
By convention, we say that the empty set is a 0-separable product set. Remark 3.2 As immediate consequences of this definition, we note that is given by
Miscellaneous properties of the obtained polynomials
Our purpose in this section is to gather some properties of the obtained d-OPSs, generalizing in a natural way the Hermite and Charlier polynomials ones. That means that, for Charlier type polynomials case, the properties to be established will be valid for .
d-Dimensional functional Vector
In this section, we express explicitly the d-dimensional functional vectors related to the resulting d-OPSs and the Laguerre type d-OPS given by Ben Cheikh and Douak [11].
As remarked by Douak and Maroni [23], a PS is d-orthogonal with respect to a d-dimensional functional vector if and only if it is also d-orthogonal with respect to the vector , where the functionals are the d first elements of the dual sequence associated to the
Concluding remark
In this section, we discuss the link between some obtained characterization theorems in this paper and N. Al-Salam ones. To this end, we defined the following classes of PSs.
Let ; be the class of PS having a generalized hypergeometric representation of the form (1.6) or equivalently, having explicit expression of the typeand let ; be the class of PS defined by (1.1). PutWe have .
N. Al-Salam [1] showed that
References (44)
- et al.
Connection problems via lowering operators
J. Comput. Appl. Math.
(2005) - et al.
A generalized hypergeometric d-orthogonal polynomial set
C.R. Acad. Sci. Paris
(2000) - et al.
Some discrete d-orthogonal polynomial sets
J. Comput. Appl. Math.
(2003) - et al.
d-Orthogonality via generating functions
J. Comput. Appl. Math.
(2007) The relation of the d-orthogonal polynomials to the Appell polynomials
J. Comput. Appl. Math.
(1996)- et al.
Une caractérisation des polynômes classiques de dimension d
J. Approx. Theory
(1995) - et al.
On d-orthogonal Tchebyshev polynomials I
Appl. Numer. Math.
(1997) Vector orthogonal relations. Vector QD-algorithm
J. Comput. Appl. Math.
(1987)Orthogonal polynomials of hypergeometric type
Ser. Math. Inform.
(1966)On characterization of certain set of orthogonal polynomials
Boll. Un. Mat. Italla.
(1964)
Sur les polynômes orthogonaux en rapport avec d’autres polynômes
Buletinul Societâtii Stiite din Cluj
Connection coefficients between Boas–Buck polynomial sets
J. Math. Anal. Appl.
On the classical d-orthogonal polynomials defined by certain generating functions I
Bull. Belg. Math. Soc.
On two-orthogonal polynomials related to Bateman’s -function
Math. Appl. Anal.
On the classical d-orthogonal polynomials defined by certain generating functions II
Bull. Belg. Math. Soc.
Dunkl–Appell d-orthogonal polynomials
Integral Transfer Special Funct.
Some generating functions of Laguerre and Hermite polynomials
Can. J. Math.
Characterization of certain sequences of orthogonal polynomials
Portugaliae Math.
Cited by (16)
D-orthogonality of discrete q-Hermite type polynomials
2013, Journal of Approximation TheoryD-orthogonality of Little q-Laguerre type polynomials
2011, Journal of Computational and Applied MathematicsOn d-symmetric classical d-orthogonal polynomials
2011, Journal of Computational and Applied MathematicsA characterization of Dunkl-classical d-symmetric d-orthogonal polynomials and its applications
2011, Journal of Computational and Applied MathematicsOn Askey-scheme and d-orthogonality, I: A characterization theorem
2009, Journal of Computational and Applied Mathematics