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Recurrence Relationsmedium
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Two solutions to a homogeneous second-order recurrence are ϕn(1)=2n\phi_n^{(1)} = 2^nϕn(1)​=2n and ϕn(2)=3⋅5n\phi_n^{(2)} = 3 \cdot 5^nϕn(2)​=3⋅5n. Calculate their Casoratian (discrete Wronskian) Wn=ϕn(1)ϕn+1(2)−ϕn(2)ϕn+1(1)W_n = \phi_n^{(1)} \phi_{n+1}^{(2)} - \phi_n^{(2)} \phi_{n+1}^{(1)}Wn​=ϕn(1)​ϕn+1(2)​−ϕn(2)​ϕn+1(1)​ and determine linear independence.