Theory of Linear Hahn difference equations

Abstract

Hahn introduced the dierence operator D_q,wf (t)=(f(qt+w)-f(t))/(t(q-1)+w) in 1949, where  0<q<1 and w>0  are fixed real numbers. This operator extends the classical differance operator V_wf(t) = (f(t + w)-f(t))/w as well as Jackson q- dierence operator D_qf(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,wX(t) + a₁(t)D^n-1_q,w x(t) + ...+ a_n(t)x(t) = 0: We introduce its fundamental set of solutions when the coefficients are constant and the Wronskian associated with D_q,w. 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.