On the use of the Cauchy-product in deriving the stress-energy tensor constraint for a string in light-cone coordinates
I shall be following the notation used in <PERSON> String Theory notes. In bosonic string theory, after fixing the gauge of the <PERSON> action using Diff/Weyl invariance to make the two-dimensional metric flat, we arrive at the free wave equation of motion for a string subject to additional constraints imposed onto the stress-energy tensor:
$$ T_{\alpha\beta} = \partial_\alpha X \cdot \partial_\beta X - \frac{1}{2}\eta_{\alpha\beta}\eta^{\rho\sigma} \partial_\rho X \cdot \partial_\sigma X, \quad \quad \alpha, \beta = 0, 1, \tag{1.32} $$
where the constraints require that:
$$ T_{01} = \dot{X} \cdot X' = 0 $$ $$ T_{00} = T_{11} = \frac{1}{2} \left( \dot{X}^2 + {X'}^2 \right) = 0 \tag{1.33}$$
which in light-cone coordinates are:
$$ \left( \partial_+ X \right)^2 = \left( \partial_- X \right)^2 = 0.\tag{1.38}$$
The Fourier mode expansion of the string in the light-cone coordinates give:
$$ \partial - X^\mu = \partial- X^\mu_R = \frac{\alpha'}{2} + \sqrt{\frac{\alpha'}{2}} \sum_{n \neq 0} \alpha^\mu_n e^{-in\sigma^-} = \sqrt{\frac{\alpha'}{2}} \sum_n \alpha_n^\mu e^{-in\sigma^-} $$
and similarly for the left-moving modes. 616
Substituting this into the constraint we have that:
\begin{align} \left( \partial_- X \right)^2 &= \frac{\alpha'}{2} \sum_{m, p} \alpha_m \cdot \alpha_p e^{-i(m+p)\sigma^-} \ &= \frac{\alpha'}{2} \sum_{m, n} \alpha_m \cdot \alpha_p e^{-in\sigma^-} \ &\equiv \alpha' \sum_{n} L_n e^{-in\sigma^-} \end{align}
where
$$L_n = \frac{1}{2} \sum_m \alpha_{n-m} \cdot \alpha_m\tag{1.39}$$
is the Virasoro operator. Now my questions is: how is the line
$$ \left( \partial_- X \right)^2 = \frac{\alpha'}{2} \sum_{m, p} \alpha_m \cdot \alpha_p e^{-i(m+p)\sigma^-} $$
derived? In my attempts I make use of the Cauchy product for the multiplication of the two infinite sums. However, this is only defined for all positive integers in the outer-sum and for all positive integers up to the first summation variable in the inner-sum:
$$ \left( \sum_{n=0}^\infty a_n \right) \cdot \left( \sum_{m=0}^\infty b_m \right) = \sum_{p=0}^\infty \sum_{q=0}^p a_q b_{p-q} $$
so I am struggling to see how $(\partial _- X)^2$, which has the product of two sums over all integers, can be evaluated? 262