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Roman Factorial


\begin{displaymath}
\left\lfloor{n}\right\rceil !\equiv\cases{
n! & for $n\geq 0$\cr
{(-1)^{-n-1}\over (-n-1)!} & for $n<0$.\cr}
\end{displaymath} (1)

The Roman factorial arises in the definition of the Harmonic Logarithm and Roman Coefficient. It obeys the identities
\begin{displaymath}
\left\lfloor{n}\right\rceil !=\left\lfloor{n}\right\rceil \left\lfloor{n-1}\right\rceil !
\end{displaymath} (2)


\begin{displaymath}
{\left\lfloor{n}\right\rceil !\over\left\lfloor{n-k}\right\r...
...lfloor{n-1}\right\rceil \cdots\left\lfloor{n-k+1}\right\rceil
\end{displaymath} (3)


\begin{displaymath}
\left\lfloor{n}\right\rceil !\left\lfloor{-n-1}\right\rceil !=(-1)^{n+(n<0)},
\end{displaymath} (4)

where
\begin{displaymath}
\left\lfloor{n}\right\rceil \equiv \cases{
n & for $n\not=0$\cr
1 & for $n=0$\cr}
\end{displaymath} (5)

and
\begin{displaymath}
n<0\equiv\cases{
1 & for $n<0$\cr
0 & for $n\geq 0$.\cr}
\end{displaymath} (6)

See also Harmonic Logarithm, Harmonic Number, Roman Coefficient


References

Loeb, D. and Rota, G.-C. ``Formal Power Series of Logarithmic Type.'' Advances Math. 75, 1-118, 1989.

Roman, S. ``The Logarithmic Binomial Formula.'' Amer. Math. Monthly 99, 641-648, 1992.




© 1996-9 Eric W. Weisstein
1999-05-25