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Multiple Laguerre polynomials: combinatorial model and Stieltjes moment representation. (English) Zbl 1509.33011

Let \(r\geq 1\) be a fixed integer; the author then introduces for \(\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_r),\ \mathbf{n}=(n_1,\dots n_r)\) the multiple Laguerre polynomials of the first kind of type II using the definition given by Mourad E. H. Ismael [M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable. With two chapters by Walter Van Assche. Cambridge: Cambridge University Press (2005; Zbl 1082.42016)] \[ \mathbf{L}_{\mathbf{n}}^{(\boldsymbol{\alpha})}=(-1)^{|\mathbf{n}|}\left(\prod_{i=1}^r (\alpha_i+1)^{\overline{n_i}}\right) e^x {}_rF_r\left(\left.\begin{matrix}(\alpha+1+n_1,\ldots,\alpha_r+1+n_r\\ \alpha_1+1,\ldots,\alpha_r+1\end{matrix}\right|-x\right), \] where \(|\mathbf{n}|=n_1+\cdots +n_r\).
In the introduction the author states that the “purpose of the present paper is twofold: (a) to give a combinatorial interpretation of the multiple Laguerre polynomials …and (b) to give an explicit integral representation for these polynomials, showing that they form a multidimensional Stieltjes moment seqience whenever \(x\leq 0\)”.
The main results are gievn in two theorems, stated below.

Theorem 2.1. The monic unsigned multiple Laguerre polynomials \[ \mathcal{L}_{\mathbf{n}}(x)=(-1)^{|\mathbf{n}|} \mathbf{L}_{\mathbf{n}}^{(\boldsymbol{\alpha})}(-x) \] have the combinatorial representation \[ \mathcal{L}_{\mathbf{n}}=\sum_{G\in\mathbf{LD}_{\mathbf{n}}}\,x^{\mathrm{pa}(G))} \prod_{i=1}^r(\alpha_i+1)^{\mathrm{cyc}_i(G)} \] (here \(\mathbf{LD}_{\mathbf{n}}\) is the set of Laguerre digraphs of the form \((V_{\mathbf{n}},A)\) with \(A\subset \vec{E}_{\mathbf{n}}\)).

Theorem 3.1. Let \(\alpha_1,\ldots,\alpha_r\geq -1\) and \(x\geq 0\). Then the multisequence \((\mathcal{L}_{\mathbf{n}}(x))_{\mathbf{n}\in \mathbb{N}^r}\) of monic unsigned multiple Laguerre polynomials is a multidimensional Stieltjes moment sequence: that is, there exists a positive measure \(\mu_{\boldsymbol{\alpha},x}\) on \([0,\infty)^r\) such that \[ \mathcal{L}_{\mathbf{n}}(x)=\int_{[0,\infty)^r},\mathbf{y}^{\mathbf{n}}d\mu_{\boldsymbol{\alpha},x}(\mathbf{y}) \] for all \(\mathbf{n}\in\mathbb{N}^r\), where \(\mathbf{y}^{\mathbf{n}}=\prod_{i=1}^r\,y_i^{n_i}\). In fact, for \(\alpha_1,\ldots,\alpha_r>-1\) we have the explicit formula \[ d\mu_{\boldsymbol{\alpha},x}(\mathbf{y})=e^{-x}{}_0F_r\left(\left.\begin{matrix} - \\ \alpha_1+1,\ldots,\alpha_r+1\end{matrix}\right|xy_1\cdots y_r\right) \prod_{i=1}^r\,\frac{1}{\Gamma(\alpha_1+1)} y_i^{\alpha_i}e^{-y_i}dy_i. \]

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Citations:

Zbl 1082.42016
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References:

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