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Roman Factorial

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|_n]!={n! for n>=0; ((-1)^(-n-1))/((-n-1)!) for n<0.
(1)

The Roman factorial arises in the definition of the harmonic logarithm and Roman coefficient. It obeys the identities

|_n]!=|_n]|_n-1]!
(2)
(|_n]!)/(|_n-k]!)=|_n]|_n-1]...|_n-k+1]
(3)
|_n]!|_-n-1]!=(-1)^(n+(n<0)),
(4)

where

|_n]={n for n!=0; 1 for n=0
(5)

and

n<0={1 for n<0; 0 for n>=0.
(6)

See also

Harmonic Logarithm, Harmonic Number, Roman Coefficient

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References

Loeb, D. E. "A Generalization of the Binomial Coefficients." 9 Feb 1995. https://arxiv.org/abs/math/9502218.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.

Referenced on Wolfram|Alpha

Roman Factorial

Cite this as:

Weisstein, Eric W. "Roman Factorial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RomanFactorial.html

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