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I have a function of the form: $f(n) = \prod_{i=1}^{n-1} (1-ai)$ Here, $a \geq 0$ and $(a*i) < 1$. For $n > 10^5$ or $10^6$, what is the best possible analytic approximation for $f(n)$ that will allow me compute values for the function with reasonable computational resources? What bounds on the error of the approximation can I expect? |
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$$f(n) = \dfrac{(-a)^{n-1} \Gamma(n - 1/a)}{\Gamma(1 - 1/a)}$$ If $a n < 1$, you'll want to use the reflection principle $$\Gamma(1-z) \; \Gamma(z) = {\pi \over \sin{(\pi z)}}$$ so $$ f(n) = \dfrac{a^{n-1} \Gamma(1/a)}{\Gamma(1-n+1/a)}$$ Use Stirling's series for the asymptotic approximation of $\ln(\Gamma(1-n+ 1/a))$, or one of its variants that you can find at http://en.wikipedia.org/wiki/Stirling%27s_approximation |
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