Shift of summation variable in sum over binomial coefficients.

Question

My question refers to the following equations:

(The summation over $\gamma$ is described here: Summation over a product of binomial coefficients)

When I perform the substitution $\beta=\alpha-\lambda+\mu+\sigma$, write everything as factorials and collect the terms again I arrive at eq. (3.13), except that my summation starts from $\beta=\sigma+\mu-\lambda$.

How do they end up with the sum starting from $\beta=0$?

(Remarks: all variables are integers, $\sigma,\lambda,\mu\geq 0$, $\lambda\geq\mu$)

Answer 1

The factor $$\binom{\lambda + \beta - \mu}{\sigma}$$ is $0$ for $\beta < \sigma + \mu - \lambda$, so the sum is the same, whether you start at $0$ or at $\sigma + \mu - \lambda$. Taking $0$ leads to simpler typography.