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Prove that $$x^{(n)} = x(x+1)(x+2)\cdots(x+n-1) = \sum_{k=0}^n \left[n\atop k\right]x^k$$ Or $$\sum_{k=0}^n \left[n\atop k\right]x^k = \frac{\Gamma(x+n)}{\Gamma(x)}$$ $$\textit{proof}$$ By induction for $n=0$ we have $$\left[0\atop 0\right] = 1$$ Assume it is true for $n$ then for $… Continue reading Posted in Striling numbers of first kind | | Leave a comment Relation between harmonic numbers and Stirling numbers of the first kind Prove that \left[n\atop 2\right] = H_{n-1}\Gamma(n) By induction on \( n$ we have for $n=2$ $$\left[2\atop 2\right] = H_{1}\times\Gamma(1) = 1$$ Assume that $$\left[k\atop 2\right] = H_{k-1}\Gamma(k)$$ Then by the recurrence relation $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading | Tagged , , , , , | Leave a comment Stirling numbers of the first kind special values proof Prove that$$\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

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