# Tag Archives: values

Prove that $$\left[n\atop 1\right] = \Gamma(n)$$ Use the recurrence relation $$\left[n+1\atop k\right] = n\left[n\atop k\right] + \left[n\atop k-1\right]$$ This implies that for $k=1$ $$\left[n+1\atop 1\right] = n\left[n\atop 1\right] + \left[n\atop0\right]$$ Now use that $\left[n\atop 0\right] = 0 … Continue reading Posted in Striling numbers of first kind | Tagged , , , , , , | Leave a comment ## Dilogarithm at 2 \mathrm{Li}_2\left(\frac{1}{2}\right)= \frac{\pi^2}{12}-\frac{1}{2}\log^2 \left(\frac{1}{2}\right)   proof  Using the duplication formula proved here \mathrm{Li}_2\left(z\right)+\mathrm{Li}_2(1-z)\, = \frac{\pi^2}{6}-\log(z) \log(1-z) \,\,  We can easily deduce that for \( z=\frac{1}{2}$ $$2\mathrm{Li}_2\left(\frac{1}{2}\right)= \frac{\pi^2}{6}-\log^2\left(\frac{1}{2}\right) \,\,$$ It follows by dividing by 2.

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