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Relation between harmonic numbers and Stirling numbers of the first kind
Prove that $$\left[n\atop 2\right] = H_{n1}\Gamma(n)$$ By induction on \( n\) we have for \( n=2\) $$\left[2\atop 2\right] = H_{1}\times\Gamma(1) = 1$$ Assume that $$\left[k\atop 2\right] = H_{k1}\Gamma(k)$$ Then by the recurrence relation $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading
Posted in Harmonic numbers, Striling numbers of first kind
Tagged Harmonic, number, numbers, proof, relation, stirling
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Stirling numbers of the first kind special values proof
Prove that $$\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 k1\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 first, kind, numbers, proof, special, stirling, values
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