
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: generating
Stirling numbers of first kind generating function
Prove the following $$\sum_{n=k}^\infty(1)^{nk}\left[n\atop k\right] \frac{z^n}{n!} = \frac{\log^k(1+z)}{k!}$$ $$\textit{proof}$$ We start by the following $$(1+z)^u = \sum_{n=0}^\infty {u \choose n} z^n$$ Now use that $${u \choose n} = \frac{\Gamma(u+1)}{\Gamma(un+1)n!}$$ Now use that $$\frac{\Gamma(u+1)}{\Gamma(un+1)} = \frac{u(u1)\cdots (un+1)\Gamma(u+1)}{\Gamma(u+1)} = (u)_n$$ This implies … Continue reading →
Posted in Signed Stirling numbers of first kind, Striling numbers of first kind

Tagged first, formula, function, generating, kind, proof, stirling

Leave a comment
Signed Stirling numbers of first kind as coefficients
Signed Stirling numbers of the first kind We define the following $$s(n,k) = (1)^{nk} \left[n\atop k\right]$$ Prove the following $$(x)_n = x(x1)(x2)\cdots (xn+1) = \sum_{k=0}^n s(n,k)x^k$$ $$\textit{proof}$$ We already proved that $$x^{(n)} = \sum_{k=0}^n \left[n\atop k\right] x^k$$ Which can be … Continue reading →
Posted in Signed Stirling numbers of first kind, Striling numbers of first kind

Tagged first, formula, generating, kind, proof, Signed, stirling

Leave a comment
Generating number of the Stirling numbers of the first kind
Prove that $$x^{(n)} = x(x+1)(x+2)\cdots(x+n1) = \sum_{k=0}^n \left[n\atop k\right]x^k$$ Or $$ \sum_{k=0}^n \left[n\atop k\right]x^k = \frac{\Gamma(x+n)}{\Gamma(x)}$$ $$\textit{proof}$$ By induction for \( n=0\) we have $$ \left[0\atop 0\right] = 1$$ Assume it is true for \( n \) then for \( … Continue reading →
Posted in Striling numbers of first kind

Tagged first, function, generating, kind, number, numbers, proof, relation, stirling

Leave a comment