Asymptotic iteration method for solving Hahn difference equations

Abstract

Abstract Hahn’s difference operator D q ; w f ( x ) = ( f ( q x + w ) − f ( x ) ) / ( ( q − 1 ) x + w ) $D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w})$ , q ∈ ( 0 , 1 ) $q\in (0,1)$ , w > 0 $w>0$ , x ≠ w / ( 1 − q ) $x\neq w/(1-q)$ 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 ) $(q;w)$ -hypergeometric equation.

Keywords

• Hahn operator
• Linear difference equations
• q-difference equations
• Polynomial solutions
• Eigenvalue problems