Scaled asymptotics for \(q\)-orthogonal polynomials
The \(q^{-1}\)-Hermite polynomials, the Stieltjes-Wigert polynomials and the \(q\)-Laguerre polynomials are typical cases of polynomials which satisfy three-term recurrence relations whose coefficients grow exponentially. The authors summarize results concerning scaled asymptotics for the above \(q\)-orthogonal polynomials. For \(0<q<1\), \(t>0\) and \(u\) a complex number, \(u\neq 0\), their scaling for the \(q^{-1}\) -Hermite polynomials is \(x_{n}(t,u)=(q^{-nt}u-q^{nt}/u)/2\) and for Stieltjes-Wigert polynomials and the q-Laguerre polynomials is \(x_{n}(t,u)=q^{-nt}u\). Their bulk asymptotics involve theta functions and exhibit a chaotic behavior when \(t\) is irrational and periodic behavior when \(t\) is rational.
scientific article; zbMATH DE number 5130334