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Nonlinear euler sum proof using stirling numbers of the first kind
Prove that $$\sum_{k=1}^\infty \frac{(H_k)^2}{k^2} = \frac{17\pi^4}{360}$$ $$\textit{proof}$$ Start by the following which can be proved by induction $$\frac{\left[n\atop 3\right]}{n!} =\frac{ (H_{n1})^2H^{(2)}_{n1}}{2n}$$ And the generating function proved here $$\sum_{n=3}^\infty \left[n\atop 3\right] \frac{z^n}{n!} = \frac{\log^3(1z)}{6}$$ Hence we get $$\sum_{n=3}^\infty ( H^{(2)}_{n1} (H_{n1})^2) … Continue reading
Stirling numbers of the first kind special values proof
Prove that $$\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
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Tagged first, kind, numbers, proof, special, stirling, values
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