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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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