## Eine Verallgemeinerung der Gleichung $$(n+1)!=n!(n+1)$$ und zugehörige vermutete Ungleichungen. (A generalization of the equation $$(n+1)!=n!(n+1)$$ and related supposed inequalities).(German)Zbl 0532.33001

For m,n,k integer with $$0\leq k\leq m-1,\quad n\geq 0$$ the equation $(m+n)!=\frac{m!}{k!(m-k-1)!}\sum^{n}_{i=0}\left( \begin{matrix} n\\ i\end{matrix} \right)(k+i)!(m+n-k-1-i)!$ is proved. By reference to considerations of the mathematical statistics the following and other similar conjectured inequalities are motivated: $\frac{m!}{k!(m-k- 1)!}\sum^{n}_{i=\ell}\left( \begin{matrix} n\\ i\end{matrix} \right)\frac{(k+i)!(m+n-k-1-i)!}{(m+n)!}>\frac{1}{2}$ for k,$$\ell,m,n$$ integer, $$0\leq k\leq m-1,\quad \ell \geq 0,\quad n\geq 0,\quad k:m=\ell:n.$$

### MSC:

 33B15 Gamma, beta and polygamma functions 26D15 Inequalities for sums, series and integrals
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### References:

 [1] Gröbner, W., Hofreiter, N.: Integraltafel, 2. Teil. Wien-New York: Springer. 1950. [2] Vietoris, L.: Vergleich unbekannter Mittelwerte auf Grund von Versuchsreihen I. Sitz.ber. Österr. Akad. Wiss.188, 329–341 (1979). · Zbl 0466.62025
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