Why cant factorials be negative

Is there a continuous function that allows us to "join the dots" and define "Factorial" for any non-negative Real number? Unfortunately this means that Gamma t is not defined when t is zero or a negative integer. The Gamma function has a simple pole at 0 and negative integers. This is named after a mathematician S. Roman, not the Romans and is used to provide a convenient notation for the coefficients of the harmonic logarithm.

Why do factorials not exist for negative numbers? Dec 26, I have a more involved discussion in a hobby-treatize about the Eulerian-triangle here. My question was intended somewhat along the line: Assume the Gamma function is not yet invented and Goldbach asks you the question: "What is -n! What would you answer? I will give my answer in this sens. What about saying the Bell numbers are the factorial numbers at negative integers? Is the answer encoded in one of the most important triangles in combinatorics?

See what Knuth says about the origin of this duality table on page I am not sure why it should be a negative infinity. Possibly because zero can be very small negative number as well as positive. I cannot derive the sign. But, I can prove that other integer negatives are also infinities. Sign up to join this community. The best answers are voted up and rise to the top.

The factorial of -1, -2, -3, Ask Question. Asked 11 years, 10 months ago. Active 3 years, 8 months ago. Viewed k times. So the question is: How could a sensible generalization of the factorial for negative integers look like? Improve this question. Bruce Arnold Bruce Arnold 1 1 gold badge 8 8 silver badges 14 14 bronze badges. When such an elementary question would be asked nowadays, it would get closed almost immediately.

Akiva Weinberger Akiva Weinberger Why do we need to look at the previous state? The -t state. What happened in the past is of no use to us.

The reference should be zero always. Also: made a slight edit. Do you see what I see? Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The factorials of real and imaginary numbers thus defined show uniformity in magnitude and satisfy the basic factorial equation c n n! Another lacuna in the existing Eularian concept of factorials is that although the factorials of negative integers are not defined, the double factorial of any negative odd integer may be defined, e.

Wikipedia b. Another strange behaviour of double factorials is that as an empty product, 0!! The present concept will remove anomalies in factorials and double factorials. The present concept generalizes factorials as applicable to real and imaginary numbers, and fractional and mutifactorials. The present paper examines the Eularian concept of factorials from basic principles and gives a new concept, based on the Eularian concept for factorials of real negative and imaginary numbers.

The factorials of negative real numbers are complex numbers. At negative integers the imaginary part of complex factorials is zero, and the factorials for -1, -2, -3, -4 are -1, 2, -6, 24 respectively. The moduli of negative real number factorials and imaginary number factorials are equal to the factorials of respective real positive numbers. The present paper also provides a general definition of fractional factorials and multifactorials.

The factorials follow recurrence relations. Beta function on the real negative axis has also been redefined in the context of new concept. Draw Function Graphs — Recheronline. Google Scholar.

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Notes Theory Discrete Math , Lefort X: History of the logarithms: an example of the development of a concept in mathematics. Cornell University. Book Google Scholar. Roman S: The logarithmic binomial formula. Amer Math Month , Srinivasan GK: The gamma function: An eclectic tour. Thukral AK: Logarithms of imaginary numbers in rectangular form: A new technique.

Can J Pure Appl Sc , 8 3 Can J Pure Appl Sc , 8 2 Weistein EW: Factorial. Weistein EW: Double factorial. Weistein EW: Beta function. Wikipedia: Factorial. Wikipedia: Double Factorial.

Wikipedia: Beta function. Wolfram Research: Factorial.