Asymptotics for the ratio and the zeros of multiple Charlier polynomials (Q5891361)

The authors investigate multiple Charlier polynomials \(C_{\vec n}(x)\) which are orthogonal with respect to \(r\) Poisson distributions with parameters \(a_1,\ldots,a_r>0\) [\textit{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)]. Ratio asymptotics NEWLINE\[NEWLINE\lim_{n\to\infty,n/N\to t}\frac{C_{\vec n+\vec e_k}(Nx)}{NC_{\vec n}(Nx)}=x-tNEWLINE\]NEWLINE are obtained. The zeros of \(C_{\vec n}(Nx)\) are proven to be asymptotically uniform on \([0,t]\) when \(n,N\to \infty\) and \(n/N\to t>0.\) More interesting asymptotics are given when some of the parameters depend on \(N\) and grow together with the degree \(|\vec n|.\)