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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 $$\sum_{n=3}^\infty ( H^{(2)}_{n1} (H_{n1})^2) … Continue reading
Relation between harmonic numbers and Stirling numbers of the first kind
Prove that $$\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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