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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?

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  • $\begingroup$ 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. $\endgroup$ – Calvin Lin Feb 20 '13 at 12:22
  • $\begingroup$ Let's say that $b\neq$0, and $a, b$ are integers. $\endgroup$ – Alon Gubkin Feb 20 '13 at 12:24
  • $\begingroup$ 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. $\endgroup$ – Hagen von Eitzen Feb 20 '13 at 12:26
  • $\begingroup$ @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. $\endgroup$ – Calvin Lin Feb 20 '13 at 12:27
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In the 1700s and 1800s a lot of work (largely forgotten now, I suspect) was done on this sort of thing. You can find much of it by googling factorial in google-books, restricting the results to come from the 19th century. Here are two such items:

Thomas Tate, A Treatise on Factorial Analysis, With the Summation of Series (1845)

Alexander Tilloch, On a new method of treating factorials and figurate numbers, Philosophical Magazine 53 (1819), 412-418.

I would not worry about the fact that much of this has been done before. First, a lot of it is probably no longer very well known. Second, a lot of it is written in a style and with notation that is difficult for a modern reader to follow. Third, your primary interest should not be worrying about whether someone has done something before, but in gaining skills by working things out for yourself.

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