Parameters of a Binomial Coefficient

Question

For a binomial coefficient $$\binom ab$$ would it be correct to say the following:

1. $b$ must be either $0$ or a positive integer. i.e. $b$ cannot be negative or a fraction.

2. $a$ can be either positive or negative, and either an integer or a fraction, subject to the condition that if $a$ is a positive integer, then $a\ge b$ (otherwise the binomial coefficient is defined as zero).

This means that we can have binomial coefficients like $$\binom {-2}3=\frac {(-2)(-3)(-4)}{1\cdot 2\cdot 3}$$ $$\binom {-\frac 13}4=\frac {-\frac 13\cdot -\frac 43\cdot -\frac 73\cdot -\frac {11}3}{1\cdot 2\cdot 3\cdot 4}$$ But binomial coefficients like $$\binom 34=0$$ as $3<4 (3,4\in \Bbb{Z})$ whilst $$\binom {3}{\frac 14}$$ is not defined.

Are there any other conditions which have not been included? Does a binomial coefficient exist for numbers which are not rational?

[Note - following from comments on this question, it appears that the limitations on parameters of a binomial coefficient $$\binom ab$$ are that both $a,b$ are real.

If follows from the same definition that if $a$ is an integer less than $b$ then then $\binom ab=0$, because of the "zero crossing" in the falling factorial of $a$.]

Answer 1

From $r!:=\Gamma(r+1)$, you can define

$$\binom rs:=\frac{r!}{(r-s)!s!}$$

for any reals (but negative integers) and when either factor at the denominator is a negative integer, the expression is defined as $0$.

For instance,

$$\binom rk=\frac{r!}{(r-k)!k!}=\frac{\Gamma(r+1)}{\Gamma(r-k+1)k!}=\frac{r(r-1)\cdots(r-k+1)}{k!}=\frac{(r)_k}{k!}$$ where the numerator is a so-called falling factorial.

This allows to write the generalized binomial theorem as

$$(a+b)^r=\sum_{k=0}^\infty\binom rka^{r-k}b^k.$$