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I would like to solve the equation xx recurrence using the operator $E$, ie, $$ (E^2 + 1)u_n = \sqrt{2}\cos[(n-1)\pi/4] \quad \Rightarrow \quad u_n = \dfrac{1}{E^2 + 1}\{\sqrt{2}\cos[(n-1)\pi/4]\} $$ I'm having trouble making sense of the term $\dfrac{1}{E^2 + 1}$.

Thanks for any help.

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If $E$ is the shift operator, then $E (e^{\lambda n}) = e^{\lambda (n+1)} = e^\lambda e^{\lambda n}$, and $(E^2+1)^{-1} e^{\lambda n} = \dfrac{1}{e^{2\lambda} + 1} e^{\lambda n}$. Now express your cosine in terms of complex exponentials...

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