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Category Archives: Striling numbers of first kind
General formula for an integral involving powers of logarithms
$$\int\limits_0^1 \dfrac{\log^{m} (1+x)\log^n x}{x}\; dx = (1)^{n+1}(n!) (m)! \sum_{\{m1\}} \sum_{k=1}^\infty \frac{(1)^k}{k^{n+2}} \prod^{l’}_{j=1}\frac{(1)^{i_j}}{(i_j)!} \left( \frac{H_{k1}^{(r_j)}}{r_j}\right)^{i_j}$$ $$\textit{solution}$$ Stirling numbers of the first kind might be useful here, Consider $$m! \sum_{k=m}^\infty (1)^{km} \left[k\atop m\right] \frac{x^k}{k!} = \log^m(1+x)$$ $$\int\limits_0^1 \dfrac{\log^m (1+x)\log^n x}{x}\; dx = … Continue reading
Posted in Euler sum, Harmonic numbers, stirling, Striling numbers of first kind
Tagged alternting, Euler, general, Integral, logarithms, powers
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Nonlinear euler sum proof using stirling numbers of the first kind
Prove that $$\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 $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading
Posted in Harmonic numbers, Striling numbers of first kind
Tagged Harmonic, number, numbers, proof, relation, stirling
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Stirling numbers of the first kind special values proof
Prove that $$\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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