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I have a recurrence relation of the following form:

$x(t+1)=\alpha x(t) t-\beta t$

Could anyone point me to a resource for how to solve the above for an arbitrary initial condition, $x(0)$?

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    For any value of $\alpha,\beta$ ? – 2017-01-30
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    Preferable. Just really want to an idea of how to approach this kind of problem. But an approach for a specific $\alpha$, $\beta$, or a subset of the real numbers might be informative. – 2017-01-30
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    It is really a very complex problem. It would (I suppose) simplify for $\alpha >0$. Let me try. – 2017-01-30

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This is not an answer but just the result from a CAS.

Considering $$x_{t+1}=\alpha\, t\, x_t-\beta\, t\qquad \qquad(\text{with }\,x_\color{red}{1}=a)$$ a CAS gave me an awful result which only simplifies if $\alpha >0$. It is $$x_t=\alpha ^{t-2} (t-1) \left(a\, \alpha \, (t-2)!-e^{\frac{1}{\alpha }} \beta \,\, \Gamma \left(t-1,\frac{1}{\alpha }\right)\right)$$ where appears the incimplete gamma function.