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تسجيل

Roman Factorial

المؤلف:  Loeb, D. E.

المصدر:  "A Generalization of the Binomial Coefficients." 9 Feb 1995. http://arxiv.org/abs/math/9502218.

الجزء والصفحة:  ...

19-5-2019

2015

Roman Factorial

 

{n! for n>=0; ((-1)^(-n-1))/((-n-1)!) for n<0. " src="http://mathworld.wolfram.com/images/equations/RomanFactorial/NumberedEquation1.gif" style="height:66px; width:176px" />

(1)

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

(2)

(3)

(4)

where

{n for n!=0; 1 for n=0 " src="http://mathworld.wolfram.com/images/equations/RomanFactorial/NumberedEquation5.gif" style="height:41px; width:117px" />

(5)

and

{1 for n<0; 0 for n>=0. " src="http://mathworld.wolfram.com/images/equations/RomanFactorial/NumberedEquation6.gif" style="height:41px; width:131px" />

(6)

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REFERENCES:

Loeb, D. E. "A Generalization of the Binomial Coefficients." 9 Feb 1995. http://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.