
Recent Posts
 Integrating a cosine log integral around a semicircle contour
 Creating Difficult integrals by the residue theorem
 Proving a trigonometric integral by integrating around an ellipse in the complex plain
 Integrating a fraction of exponential and trignometric using rectangular contour
 Integrating around a triangular contour for Fresnel integral
Recent Comments
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral representation of the digamma function using Abel–Plana formula
 Ricardo on Integral representation of the digamma function using Abel–Plana formula
 Zaidalyafeai on Integral of arctan and log using contour integration
 tired on Integral of arctan and log using contour integration
Archives
Categories
Meta
Tag Archives: stirling
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 $$\left[k+1\atop 2\right] = k\left[k\atop 2\right] + \left[k\atop … Continue reading
Posted in Harmonic numbers, Striling numbers of first kind
Tagged Harmonic, number, numbers, proof, relation, stirling
Leave a comment
Stirling numbers of the first kind special values proof
Prove that $$\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 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
Leave a comment