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Category Archives: Signed Stirling numbers of first kind
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

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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

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