1

When generalizing the factorial, we often refer to the gamma function, which gives $(n-1)!$ for $\Gamma(n)$. Why do we not use the Pi function instead, which provides the factorial of the input? Is there a formatting issue with it or is it something else?

Mason
  • 3,792
Manu Sankaran
  • 21
  • 1
    There's no inherent issue with it; it's just that some formulas appear nicer with one as opposed to the other, and the gamma function is also a bit more known.

    Examples of formulas more pleasant with the gamma function are the reflection formula for $\sin(z)$,

    $$\Gamma(z) \Gamma(1-z) = \frac{\pi}{\sin(\pi z)}$$

    as opposed to

    $$\Pi(z-1) \Pi(-z) = \frac{\pi}{\sin(\pi z)} \implies \Pi(z) \Pi(-z) = \frac{\pi z}{\sin(\pi z)} = \frac{1}{\operatorname{sinc}(z)}$$

    [cont.]

     – PrincessEev May 08 '23 at 03:44
  • 1
    which does not so clearly denote a reflection. Though I suppose one could prefer the latter formula in terms of $\Pi$ since $\operatorname{sinc}(z)$ (or $\operatorname{sinc}(\pi z)$ since definitions vary) is itself useful and the formula is still aesthetically pleasing in some sense.

    Another is the definition of the beta function,

    $$\mathrm{B}(x,y) := \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)} = \frac{\Pi(x-1) \Pi(y-1)}{\Pi(x+y-1)}$$

    so in turn

    $$\frac{xy}{x+y} \mathrm{B}(x,y) = \frac{\Pi(x) \Pi(y)}{\Pi(x+y)}$$

    for which I can foresee no justification for the use of $\Pi$. [cont.]

     – PrincessEev May 08 '23 at 03:44
  • 2
    Another thought is that, since $\Pi(x)$ is meant to align to $x!$ anyways on the nonnegative integers, one could just say "define $x!$ as the analytic continuation of the usual definition that is given for $x \in \mathbb{Z}_{\ge 0}$" and just roll with that.

    Some more discussion on MathOverflow is here, and on MSE here.

     – PrincessEev May 08 '23 at 03:44

0 Answers0