# Generalization of the Factorial function

Is there any standard generalization of the Factorial function where the "skips" per multiplication is a parameter? For example, one generalization could be: $a(a-b)(a-2b)(a-3b)...1$

I tried to generalize it myself and came up with a few interesting results.

I defined $f(a, b)$ as $f(a, b) = a(a-\frac{1}{b})(a-\frac{2}{b})(a-\frac{3}{b})...$ where $b\neq0$ and $a,b$ are integers. Here are some of the results I've came up with:

1. The factorial of $n$ would be $f(n,1)$.
2. $(nk)!$ = $k^nf(n,k)$ because $(nk)!=nk(nk-1)(nk-2)(nk-3)... = k^nn(n-\frac{1}{k})(n-\frac{2}{k})(n-\frac{3}{k})$
3. $(nk)! = (kn)!$ => $k^nf(n,k)=n^kf(k,n)$ => $\frac{k^n}{n^k} = \frac{f(k,n)}{f(n, k)}$
4. From 1, 2 and 3: $f(nk, 1) = k^nf(n,k)=n^kf(k,n)$
5. $f(n, 0) = \infty$ because $f(n,0)=n(n-0)(n-0)(n-0)...=n^\infty=\infty$

If it's not standardized - would it actually be useful for anything?

• Are you restricting only to integers (not mentioned)? You likely want $\frac {1}{b}$ to be a non-zero multiple of $a$, otherwise your value is undefined. Note that you only get the factorial on integers, but not on the non-integers. In particular, check out the Gamma Function. – Calvin Lin Feb 20 '13 at 12:22
• Let's say that $b\neq$0, and $a, b$ are integers. – Alon Gubkin Feb 20 '13 at 12:24
• Looks like $f(a,b)=\frac{(ab)!}{b^a}$ whenever this makes sense. You may generalize the factorial and power in this expressionusing the Gamma function and exponentials. – Hagen von Eitzen Feb 20 '13 at 12:26
• @Alon I would encourage you to read up on the Gamma Function, which extends the factorial function to non-integer values. Note that in your function, if $a, b$ are integers then you always get an integer factorial. – Calvin Lin Feb 20 '13 at 12:27