<p>We propose a fractional-order discrete model to capture the dynamics of an economic adoption process with indirect interactions and short-term memory effects. Using a truncated Grünwald-Letnikov (GL) fractional-difference operator of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\alpha \in (0,1]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> over a finite horizon of <i>k</i> past steps, the model incorporates memory in a computationally tractable way. A three-dimensional fractional system is introduced, describing non-adopters, adopters, and accumulated knowledge spillovers, analogous to indirect transmission in epidemic models. Sufficient conditions for nonnegativity and boundedness of the trajectories are derived to ensure economic feasibility. An <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\alpha ,k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-Basic Adoption Number is defined to analyze the local stability of equilibria, and a lifted companion-matrix approach is employed to relate stability to the spectral radius, revealing how <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation> and <i>k</i> affect the speed of adjustment and adoption dynamics. Numerical simulations illustrate the impact of fractional memory on technology diffusion, highlighting the persistence effects of knowledge spillovers in economic adoption.</p>