Timeline for Why is the Gamma function shifted from the factorial by 1?
Current License: CC BY-SA 3.0
8 events
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| Mar 14, 2020 at 5:07 | comment | added | Yemon Choi | +1 (10 years after the fact) for the convolution semigroup observation - this semigroup property seems to be quite useful for some work I've recently been doing on positive definite functions on the "ax+b group" of the reals. | |
| Mar 12, 2020 at 14:02 | comment | added | Novice C | @PietroMajer while I also cannot follow how one arrives at a binomial coefficient, it hardly corroborates that the Beta function implies Gamma is superior. The point still stands that a modified Beta function could easily be written in terms of the Pi function—in parallel to the example you have given. If you want to argue the Beta function we have is more natural than the modified Beta function that corresponds to Pi you can, but just reduces to a weaker argument in this case of saying Gamma is more natural than Pi. | |
| Jan 30, 2016 at 18:55 | comment | added | Pietro Majer | @Stephen Montgomery-Smith. In fact the formula you wrote is wrong, corroborating my claim that $\Gamma(t)\Gamma(s)/\Gamma(t+s)$ is easier to deal with ;) | |
| Jul 16, 2015 at 1:53 | comment | added | Tom Copeland | $\int_0^z \frac{x^{t-1}}{(t-1)!} \frac{(z-x)^{s-1}}{(s-1)!} \, dx = D_{z}^{-s} \frac{z^{t-1}}{(t-1)!} = \frac{z^{s+t-1}}{(s+t-1)!}$ is easy to remember if one can remember to associate $s=1$ with integration and $s=0$ with the identity or Dirac delta function. | |
| Jul 16, 2015 at 0:09 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Jul 15, 2015 at 21:00 | comment | added | Stephen Montgomery-Smith | You would define $\tilde B(t,s) = \int_0^1 x^t (1-x)^s \, dx$. Then $\tilde B(m,n) = 1\big/ \binom{m+n}m$. | |
| May 26, 2010 at 7:29 | history | edited | Pietro Majer | CC BY-SA 2.5 |
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| May 26, 2010 at 7:23 | history | answered | Pietro Majer | CC BY-SA 2.5 |