Talk:Factorial

message: An imperative way of calculating factorial may be more understandable. In Python code:
def fact(x):
    result = 1
    while x > 1:
        result *= x   # multiply result by x
        x -= 1        # decrease x by 1
    return result
and in C:
// function factorial
int fac(int fc)
   {
   int i, ret = 1;
   for (i = 1; i<=fc; i++)
      {
      ret*= i;
      }
   return (ret);
   }
Moved from the article. Is this really worth mention? -- Taku 21:38, 25 April 2003 (UTC)
thread_title:  Moved from article 

message: The sum of the reciprocals of the sum of the first n integers (the Triangular numbers) is 2. The sum of the reciprocals of the sum of the first n Triangular numbers (the Tetrahedral numbers) is 1.5.
The sum of the reciprocals of the product of the first n integers (the Factorials) is e, so I wonder, is the sum of the reciprocals of the product of the first n Factorials (the Superfactorials), which is 1.5868056, or the sum of the reciprocals of the sum of the first n Factorials which is 1.47608642, expressible in terms of e?— Preceding unsigned comment added by Robo37 (talk • contribs) 20:01, 12 August 2011 (UTC)
thread_title:  Hmmm.... 

message: The statement "There are infinitely many ways to extend the factorials to a continuous function." is true, but trivial and rather useless. As the cited source states,  This is of course true of any set of discrete points. The interesting part is that there are infinite ways to do it within certain confines, most importantly while still satisfying the recurrence relation. I therefore added the qualifier "that satisfies the recurrence relation  for non-integer values", which was reverted on the basis that it would exclude an interpolation based on Hadamard's gamma function. That, to me, misses the point. It also excludes interpolating the points linearly or indeed arbitrarily. I suppose it would be possible to rephrase it to make both statements at once (e.g. "There are infinitely many ways to extend the factorials to a continuous function, which remains true if the resulting function  is required to satisfy the recurrence relation  for non-integer values."), but I don't think the bare statements that we have now is satisfactory. TompaDompa (talk) 11:36, 19 August 2023 (UTC)
replies: text: The point is that you have to constrain things somehow to be able to say that Gamma is the canonical interpolation. Your edit adds half the constraint, turning that thought from "you have to constrain things" to "we have already constrained things but you have to constrain more things". What is so natural about that choice? Why not instead start with the other half of the Bohr–Mollerup theorem, and only consider log-convex functions? Or why not leave the constraints out of it until they are needed for uniqueness? I'm not embarrassed to say trivial things when they're relevant. Not every statement in our mathematics articles has to have deep reasoning behind it. —David Eppstein (talk) 13:02, 19 August 2023 (UTC)
replies: text: The recurrence relation is a fundamental and defining property of factorials—it's what makes factorials factorials, so to speak. When I first heard of interpolating factorials many years ago, I took for granted that the recurrence relation would hold for the non-integer relations, because I thought that without that property it wouldn't be much of an interpolating function. The non-uniqueness of the Gamma function in this regard is, I think, very important. TompaDompa (talk) 16:07, 19 August 2023 (UTC)
replies: text: Hadamard's Γ obeys a form of the recurrence relation. But it is a form with an extra term that happens to be zero on the positive integers. —David Eppstein (talk) 07:35, 20 August 2023 (UTC)
replies: text: I must admit that I don't understand what you're getting at. TompaDompa (talk) 07:53, 20 August 2023 (UTC)
replies: text: Hadamard's gamma function#Properties. When you generalize from integers to reals, it may be the case that part of a formula that vanish for integers becomes visible for reals. So although I agree that the usual Γ is usually the correct interpolation, I don't see the rationale for insisting that only functions obeying the integer version of the recurrence can be of any interest. —David Eppstein (talk) 09:15, 20 August 2023 (UTC)
replies: text: That's not what I'm saying. What I'm saying is way closer to the Gamma function not being the "correct" interpolation—or perhaps even more to the point not the correct interpolation. Had there been infinitely many interpolations but only one satisfying the recurrence relation for non-integer values, it may very well have been the case that many interpolations are interesting for one reason or another but the only one with that property might be considered the "correct" extension to non-integer values. But that's not the case (and what I wanted to make clearer to readers). The Gamma function is the most commonly used interpolation because it has useful properties, but using it to extend the factorials to non-integer values, in general, is convention rather than correct. TompaDompa (talk) 17:20, 20 August 2023 (UTC)
thread_title:  Infinitely many continuous extensions 

message: I couldn't find anything about the factorial of the imaginary unit on this page. Adding the factorial of the imaginary unit can be quite useful. Bera678 (talk) 16:18, 21 December 2023 (UTC)
replies: text: You probably could find a section titled "Continuous interpolation and non-integer generalization" and headed by "Main article: Gamma function". That suggests that if this material is anywhere it should be in the Gamma function article. However, I did not see anything about \Gamma(i+1) in Gamma function#Particular values. Is there any reason to think that i+1 has any special meaning as a parameter of the Gamma function, making it significant enough to report its value in that article? —David Eppstein (talk) 16:58, 21 December 2023 (UTC)
thread_title:  Adding factoriel of 'imaginary unit'