  In this paper, we provide a complete Plancherel-Rotach asymptotic analysis of
polynomials that satisfy a second-order difference equation with linear
coefficients. According to the signs of the parameters, we classify the
difference equations into six cases and derive explicit asymptotic formulas of
the polynomials in the outer and oscillatory regions, respectively. It is
remarkable that the zero distributions of the polynomials may locate on the
imaginary line or even on a sideways Y-shape curve in some cases.
