The Laguerre polynomials in several variables (Q2843886)

Suppose that for each \(j\) with \(1\leq j\leq r\) \(\bigl \{p_{n}^{(j)}(x)\:n\geq 0\bigr \}\) is a family of orthogonal polynomials for the weight function \(w^{(j)}(x)\) defined on a possibly infinite interval \((a,b)\), then the polynomials \(\prod_{j=1}^{r}p_{n_{j}}^{(j)}(x_{j})\) (for \((n_{1},\dots ,n_{r})\in \mathbb {Z}_{\geq 0}^{r})\) are orthogonal for the weight function \(\prod_{j=1}^{r}w^{(j)}(x_{j})\) on \((a,b)^{r}\subset \mathbb {R}^{r}\). This paper concerns these structures composed of Laguerre polynomials \(L_{n}^{(\alpha)}\) (with weight \(x^{\alpha}e^{-x}\) on \((0,\infty)\)) and Hermite polynomials \(H_{n}\) (with weight \(e^{-x^{2}}\) on \((-\infty ,\infty)\)). The relations \(H_{2n}(x)=(-1)^{n}2^{2n}n!L_{n}^{(-1/2)}(x^{2})\) and \(H_{2n+1}(x)=(-1)^{n}2^{2n+1}n!xL_{n}^{(1/2)}(x^{2})\) are displayed in the product setting. Appropriate products of the well-known generating functions for the one-variable polynomials are used to generate the product polynomials.NEWLINENEWLINETwo families of polynomials \({h_{n}},{l_{n}}\) are used in various identities that could be considered as generating functions. For brevity let \(\alpha ,\lambda ,\mu \) denote the multi-indices NEWLINE\[NEWLINE (\alpha_{j})_{j=1}^{r},\;(\lambda_{j})_{j=1}^{r},\;(\mu_{j})_{j=1}^{r}, NEWLINE\]NEWLINE respectively. Then \(h_{n}^{(\alpha)}(x)\) is defined to be the coefficient of \(t^{n}\) in \(\prod_{j=1}^{r}(1-x_{j}t^{j})^{-\alpha_{j}}\) and \(l_{n}^{(\alpha ,\lambda ,\mu)}(x)\) is the coefficient of \(t^{n}\) in NEWLINE\[NEWLINE {\displaystyle \prod \limits_{j=1}^{r}}\left \{{\displaystyle \sum \limits_{n_{j}=0}^{\infty}} \dfrac {\left (\lambda_{j}\right)_{n_{j}}}{\left (\mu_{j}\right)_{n_{j}}}L_{n_{j}}^{\left (\alpha_{j}\right)}\left (x_{j}\right)t^{n_{j}}\right \}. NEWLINE\]NEWLINE (the authors label \(l_{n}(x)\) only by \(\alpha \)).