**Looking for a difference that makes a difference.**

**Flipping the gamma function and looking at Newton interpolation provides another angle on the question:**

Accepting the general binomial coefficient as a natural extension of the integral coefficient through the Taylor series, or binomial theorem, for $(1+x)^{s-1}$ and with the finite differences $\bigtriangledown^{s-1}_{n}c_n=\sum_{n=0}^{\infty}(-1)^n \binom{s-1}{n}c_n$, Newton interpolation gives
$$\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j} \frac{x^j}{j!}=\frac{x^{s-1}}{(s-1)!}$$

for $Real(s)>0,$ so the values of the factorial at the nonnegative integers determine uniquely the generalized factorial, or gamma function, in the right-half of the complex plane through Newton interpolation.
 
Then with the sequence $a_j=1=\int_0^\infty e^{-x} \frac{x^{j}}{j!}dx$,

$\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}a_j=\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}1=1=\bigtriangledown^{s-1}_{n} \bigtriangledown^{n}_{j}\int_0^\infty e^{-x} \frac{x^{j}}{j!}dx=\int_0^\infty e^{-x} \frac{x^{s-1}}{(s-1)!}dx$.

This last integral allows the interpolation of the gamma function to be analytically continued to the left half-plane as in [MSE-Q132727][1], so the factorial can be uniquely extended from it's values at the non-negative integers to the entire complex domain as a meromorphic function. 

This generalizes in a natural way; the integral, a modified Mellin transform, can be regarded as a means to interpolate the coefficients of an exponential generating function (in this particular case, exp(x)) and extrapolate the interpolation to the whole complex plane. (See my examples in [MO-Q79868][2] and [MSE-Q32692][3] and try the same with the exponential generating function of the Bernoulli numbers to obtain an interpolation to the Riemann zeta function.) 

From these perspectives--its dual roles as a natural interpolation of a sequence itself and in interpolating others and also as an iconic meromorphic function--the gamma function provides the "best" generalization of the factorial from the nonneg integers to the complex plane.


**To more sharply connect *pbrooks* interest in fractional calculus and the gamma function with *Quiaochu's* in the beta integral, it's better to look at a natural sinc interpolation of the binomial coefficient:** 

Consider the fractional integro-derivative 
 
$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi i}\displaystyle\oint_{|z-x|=|x|}\frac{z^{\alpha}}{\alpha!}\frac{\beta!}{(z-x)^{\beta+1}}dz=FP\displaystyle\int_{0}^{x}\frac{z^{\alpha}}{\alpha!}\frac{(x-z)^{-\beta-1}}{(-\beta-1)!} dz$

$=\displaystyle\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$ 

where FP denotes a Hadamard-type finite part, $x>0$, and $\alpha$ and $\beta$ are real.

For $\alpha>0$ and $\beta<0$, the finite part is not required for the beta integral, and it can be written as

$\displaystyle\int_{0}^{1}\frac{(1-z)^{\alpha}}{\alpha!}\frac{z^{-\beta-1}}{(-\beta-1)!} dz=\sum_{n=0}^{\infty } (-1)^n
\binom{\alpha }{n}\frac{1}{n-\beta}\frac{1}{\alpha!}\frac{1}{(-\beta-1)!}$

$=\displaystyle\sum_{n=0}^{\infty }\frac{\beta!}{(\alpha-n)! n!}\frac{\sin (\pi (\beta -n))}{\pi (\beta -n)}=\frac{1}{(\alpha-\beta)!}, $ or

$$\displaystyle\sum_{n=0}^{\infty }\frac{1}{(\alpha-n)! n!}\frac{\sin (\pi (\beta -n))}{\pi (\beta -n)}=\frac{1}{(\alpha-\beta)!\beta!},$$ 

where use has been made of $\frac{\sin (\pi \beta)}{\pi \beta}=\frac{1}{\beta!(-\beta)!}$, and $\alpha$, of course, can be a positive integer. The final sinc fct. interpolation holds for $Real(\alpha)>-1$ and all complex $\beta$.

**Euler's motivation** (update July 2014):

R. Hilfer on pg. 18 of "[Threefold Introduction to Fractional Derivatives][4]" states, "Derivatives of non-integer (fractional) order motivated Euler to introduce the Gamma function ...." Euler introduced in the same reference given by Hilfer essentially 

$\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=\frac{x^{\alpha-\beta}}{(\alpha-\beta)!}$.


 [1]: http://math.stackexchange.com/questions/13956/domain-of-the-gamma-function/132727#132727
 [2]: http://mathoverflow.net/questions/79868/what-does-mellin-inversion-really-mean/79925#79925
 [3]: http://math.stackexchange.com/questions/32692/explicitly-reconstructing-a-function-from-its-moments/128935#128935
 [4]: http://www.icp.uni-stuttgart.de/~hilfer/publikationen/pdfo/ZZ-2008-ATFaA-17.pdf