- #1
MAGNIBORO
- 106
- 26
hi, i was thinking that every function that satisfies the conditions
$$f(0)=1$$
$$f(n+1)=(n+1)f(n)$$
could be a generalization of the factorial function, and why the gamma function is the only function that complies with this conditions?
I mean why don't exist other functions, or functions based of integrals that also complies with the 2 requirements?
Even I can obviate the first property and we get a "dephased" factorial function (like gamma function) so every function that complies with the property $$f(n+1)=(n+1)f(n)$$ could be an extension of the factorial, why the UNIQUE function that make that is the gamma?
Thanks
$$f(0)=1$$
$$f(n+1)=(n+1)f(n)$$
could be a generalization of the factorial function, and why the gamma function is the only function that complies with this conditions?
I mean why don't exist other functions, or functions based of integrals that also complies with the 2 requirements?
Even I can obviate the first property and we get a "dephased" factorial function (like gamma function) so every function that complies with the property $$f(n+1)=(n+1)f(n)$$ could be an extension of the factorial, why the UNIQUE function that make that is the gamma?
Thanks