Structure relations of classical multiple orthogonal polynomials by a generating function (Q2861456)

The author continues his study of multiple orthogonal polynomial systems (OPS), building on his results in [Integral Transforms Spec. Funct. 18, No. 12, 855--869 (2007; Zbl 1134.33007)] and [in: Proceedings of the 5th Asian mathematical conference (AMC), June 22--26, 2009, Kuala Lumpur, Malaysia. Vol. III: Statistics, operations research and miscellaneous. Penang: Universiti Sains Malaysia, School of Mathematical Sciences. 44--51 (2009; Zbl 1207.33017)].NEWLINENEWLINENEWLINEWith \(\vec{n}=(n_1,n_2,\dots,n_r)\in \mathbb{N}_0^r\) (\(r\geq 2)\), the OPS \(\{P_{\vec{n}}(x)\}_{|\vec{n}=0}^{\infty}\) is defined by {\parindent=8mm \begin{itemize}\item[(i)] deg\(\,P_{\vec{n}}=|\vec{n}|=n_1+\cdots +n_r\), NEWLINE\item[(ii)] there exist \(r\) positive weight functions \(w_i\), \(i=1,2,\dots,r\), with NEWLINENEWLINE\[NEWLINE\int_{-\infty}^{\infty}\,x^k P_{\vec{n}}(x)w_i(x)dx=0\;\text{ for }\; k=0,1,\dots,n_i -1.NEWLINE\]NEWLINENEWLINEHis main results concern three types of classical multiple OPSs: {\parindent=6mm \begin{itemize}\item[-] multiple Hermite polynomials (\(w_i(x)=\exp{({\delta\over 2}x^2+\alpha_ix)}\text{ on }(-\infty,\infty),~\delta<0,~\alpha_i\not=\alpha_j\text{ for }i\not= j\)), \item[-] multiple Laguerre I polynomials (\(w_i(x)=x^{\alpha_i}\exp{(\beta x)}\) on \((0,\infty)\), \(\alpha_i>-1\), \(\beta<0\), \(\alpha_i-\alpha_j\not\in\mathbb{Z}\) for \(i\not= j\)), \item[-]multiple Laguerre II polynomials (\(w_i(x)=x^{\alpha}\exp{(\beta_i x)}\) on \((0,\infty)\), \(\alpha>-1\), \(\beta_i<0\), \(\beta_i\not=\beta_j\) for \(i\not= j\)). NEWLINENEWLINE\end{itemize}} And, moreover, two other systems: {\parindent=6mm \begin{itemize}\item[-] Jacobi-Piñero polynomials (\(w_i(x)=x^{\alpha_i}(1-x)^{\alpha}\), \(i=1,2\), on \((0,1)\), \(\alpha,\alpha_1,\alpha_2>-1\), \(\alpha_1-\alpha_2\not\in \mathbb{Z}\)), \item[-] multiple Bessel polynomials (\(w_i(x)=x^{\alpha_i}\exp{(\gamma/x)}\), \(i=1,2\), on the unit circle in the complex plane\end{itemize}}, \(\alpha_1,\alpha_2>-1\), \(\gamma\neq 0\), \(\alpha_1-\alpha_2\not\in \mathbb{N}\)). NEWLINENEWLINE\end{itemize}} In each case, the generating function of the OPS, found in D.W. Lee's papers cited above, are used to derive new recurrence relations in the three classical multiple OPSs and to derive recurrence relations and a differential-difference equation for the other two OPSs.NEWLINENEWLINEThe proofs are straightforward; a nicely written paper.