Roman Factorial

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

The Roman factorial arises in the definition of the Harmonic Logarithm and Roman Coefficient. It obeys the identities

$$\left\lfloor{n}\right\rceil !=\left\lfloor{n}\right\rceil \left\lfloor{n-1}\right\rceil !$$ (2)

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

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

where

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

and

$$n<0\equiv\cases{ 1 & for $n<0$\cr 0 & for $n\geq 0$.\cr}$$ (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.