Theory of Linear Hahn Difference Equations

Abstract

Hahn introduced the dierence operator Dq;!f(t) = 􀀀 f(qt + !) 􀀀 f(t)=􀀀 t(q 􀀀 1) + ! in 1949, where 0 0 are fixed real numbers. This operator extends the classical dierence operator M! f(t) = (f(t + !) 􀀀 f(t))=! as well as Jackson q􀀀 dierence operator Dqf(t) = (f(qt)􀀀f(t))=(t(q 􀀀1)). In this paper, our target is to give a rigorous study of the theory of linear Hahn dierence equations of the form a0(t)Dn q;!x(t) + a1(t)Dn􀀀1 q;! x(t) + ::: + an(t)x(t) = 0: We introduce its fundamental set of solutions when the coecients are constant and the Wronskian associated with Dq;!. Hence, we obtain the corresponding Liouville's formula. Also, we derive solutions of the first and second order linear Hahn dierence equations with non-constant coffiecients. Finally, we present the analogues of the variation of parameter technique and the annihilator method for the non-homogeneous case.