Asymptotic iteration method for solving Hahn difference equations

Abstract

Hahn's difference operator $D_{q;w}f\left(x\right)=\left(f\right(qx+w)-f(x\left)\right)/\left(\right(q-1)x+w)$ , $q\in (0,1)$ , $w>0$ , $x\ne w/(1-q)$ is used to unify the recently established difference and q-asymptotic iteration methods (DAIM, qAIM). The technique is applied to solve the second-order linear Hahn difference equations. The necessary and sufficient conditions for polynomial solutions are derived and examined for the $(q;w)$ -hypergeometric equation.

Keywords: Eigenvalue problems; Hahn operator; Linear difference equations; Polynomial solutions; q-difference equations.