d-Orthogonality of Hermite type polynomials

doi:10.1016/j.amc.2007.11.040

Applied Mathematics and Computation

Volume 202, Issue 1, 1 August 2008, Pages 24-43

https://doi.org/10.1016/j.amc.2007.11.040 Get rights and content

Abstract

In this paper, we state a characterization theorem for d-orthogonal polynomials of Hermite type. As application, we solve two characterization problems related to Gould–Hopper polynomials and Charlier type polynomials. Various properties of the resulting generalized hypergeometric polynomials are singled out. In particularly, we derive the d-dimensional functional vectors ensuring the d-orthogonality of these polynomials. The obtained weight functions are reduced to the well-known weight functions of Charlier and Hermite polynomials for $d = 1$ . We also explicitly express the d-dimensional functional vector associated with polynomials of Laguerre type, which are related to the obtained 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 ${γ_{n, m} (k)}_{n \in N, k \in Z}$ be the sequence given by (1.3) with $N = D = 1$ . Then we have $γ_{n, m} (k) - γ_{n + 1, m} (k) = \frac{m (- n)_{m - 1}}{c^{m}} γ_{n + 1 - m, m} (k - 1), n ⩾ m, k \in Z,$ $k^{[i]} γ_{n, m} (k) = \frac{(- n)_{im}}{c^{im}} γ_{n - im, m} (k - i), i ⩽ [\frac{n}{m}], n \in N, k \in Z .$ $x^{[i]}$ being the falling factorial polynomials given by $x^{[0]} = 1 and x^{[i]} = i! (\begin{matrix} x \\ k \end{matrix}) = x (x - 1) \dots (x - i + 1), i = 1, 2, \dots .$

Proof

The relations (2.1), (2.2) can be deduced from the identities: $(- n)_{i} - (- n - 1)_{i} = i (- n)_{i - 1}, i, n \in N,$ $(a)_{i + j} = (a)_{i} (a + i)_{j}, i, j \in N . □$

Lemma 2.2

Let ${P_{n}}_{n ⩾ 0}$ 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 ${λ_{1}, \dots, λ_{s}}$ be a set of s polynomials of degree one, $s ⩾ 1$ . ${λ_{1}, \dots, λ_{s}}$ is called a s-separable product set if and only if there exists a polynomial $π$ such that $\prod_{i = 1}^{s} (λ_{i} (x) + y) = (\prod_{i = 1}^{s} λ_{i} (x)) + π (y) .$

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 $π$ is given by $π (y) = [\prod_{i = 1}^{s} (]$

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 $m = 1$ .

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 ${P_{n}}_{n ⩾ 0}$ is d-orthogonal with respect to a d-dimensional functional vector $Γ =^{t} (Γ_{0}, Γ_{1}, \dots, Γ_{d - 1})$ if and only if it is also d-orthogonal with respect to the vector $U =^{t} (u_{0}, u_{1}, \dots, u_{d - 1})$ , where the functionals $u_{0}, u_{1}, \dots, u_{d - 1}$ are the d first elements of the dual sequence ${u_{n}}_{n ⩾ 0}$ 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 $A_{m}; m ⩾ 2$ ; be the class of PS having a generalized hypergeometric representation of the form (1.6) or equivalently, having explicit expression of the type $P_{n} (x; c, m, (a_{p}), (b_{q})) = \sum_{k = 0}^{[\frac{n}{m}]} \frac{(- n)_{mk}}{c^{km} k!} \frac{{[a_{p}]}_{k}}{[b_{q}]_{k}} x^{n - mk},$ and let $B_{m}; m ⩾ 2$ ; be the class of PS defined by (1.1). Put $A = ⋃_{m ⩾ 2} A_{m} and B = ⋃_{m ⩾ 2} B_{m} .$ We have $A_{2} \subset A \subset B$ .

N. Al-Salam [1] showed that

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d-Orthogonality of Hermite type polynomials

Abstract

Section snippets

Introduction and main result

Proof of Theorem 1.1

Two characterization theorems

Miscellaneous properties of the obtained polynomials

d-Dimensional functional Vector

Concluding remark

J. Comput. Appl. Math.

C.R. Acad. Sci. Paris

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Comput. Appl. Math.

J. Approx. Theory

Appl. Numer. Math.

J. Comput. Appl. Math.

Orthogonal polynomials of hypergeometric type

Ser. Math. Inform.

On characterization of certain set of orthogonal polynomials

Boll. Un. Mat. Italla.

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 Jnu,v-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.

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