Abstract
Hahn’s difference operator D_{q;w}f(x) =({f(qx+w)-f(x)})/({(q-1)x+w}), qin (0,1), w>0, xneq 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.
Highlights
Hahn [1, 2] introduced, for x = w/(1 – q), the quantum difference operator f (q x + w) – f (x) Dq;wf (x) =, (q – 1) x + w q ∈ (0, 1), w > 0, D0q;wf (x) = f (x), Dnq;wf (x) = Dq;w Dnq;–w1f (x), n = 1, 2, . . . , (1)where f is a function defined on an interval I of R which contains w/(1 – q)
2 Preliminary definitions and properties of Hahn operator To make this paper self-contained, we review the mathematical properties of the Hahn operator [1, 2, 10, 11, 18, 29,30,31,32] and provide the proofs for other new properties developed for the present work
These two forms (38) and (40) are equivalent, we shall focus our attention on the form (40) to investigate the solutions of the second-order linear Hahn difference equation (40) with variable coefficients
Summary
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. 3, the solutions of the first- and second-order linear Hahn difference equations The necessary and sufficient conditions for the existence of polynomial solutions of the second-order linear Hahn difference equation are derived and proved.
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