Some discrete multiple orthogonal polynomials
This very interesting paper extends a section of the Askey-tableau to the situation of discrete multiple orthogonal polynomials. The attention is restricted to so called type II systems. Given \(r\) positive measures \(d\mu_j(x)\), with support on the real line, and a multi-index \({\vec n}=(n_1,n_2,\ldots,n_r)\) of natural numbers, the multiple orthogonal polynomial \(P_{\vec n}\) associated with this multi-index is a polynomial of degree at most \(n_1+n_2+\cdots n_r\) that satisfies \[ \int_{\Delta_j} P_{\vec n}(x)x^k d\mu_j(x)=0, \quad k=0,1,\dots,n_j-1,\;j=1,2,\dots,r, \] where \(\Delta_j\) is the smallest real interval containing the support of \(d\mu_j\). Their results cover the following cases (with limit relations between them): (a) Multiple Charlier: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {a_j^k \over k!}\delta(x-k) dx\), \(a_j>0\) different; (b) Multiple Meixner I: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {(\beta)_k c_j^k \over k!}\delta(x-k)) dx\), \(0<c_j<1\) different, \(\beta>0\); (c) Multiple Meixner I: \(d\mu_j(x)= (\sum_{k=0}^{\infty} {(\beta_j)_k c^k \over k!}\delta(x-k) dx\), \(\beta_j>0\) different, \(0<c<1\); (d) Multiple Kravchuk: \(d\mu_j(x)=(\sum_{k=0}^N {N\choose k}p_j^k(1-p_j^{n_k} \delta(x-k)) dx\), \(0<p_j<1\) different; (e) Multiple Hahn: \(d\mu_j(x)= (\sum_{k=0}^N {(\alpha_1+1)_k \over k!} {(\beta+1)_{N-k}\over (N-k)!}\delta(x-k)) dx\), \(\alpha_j>-1\) different, \(\beta>-1\), where \((s)_0=1\), \((s)_k=s(s+1)\cdots (s+k-1) (k\geq 1)\) and \(1\leq j\leq r\). For \(r=2\) several explicit formulae are given.
scientific article; zbMATH DE number 1915218