Cauchy series "Product Like"

Question

If I have:

$$\sum_{q=0}^{+\infty}\alpha_q \sum_{r=0}^q \beta_r\gamma_{q-r}$$

Is there a way to manipulate this expression in order to simplify it or at least rewrite into another form?

I've just noticed that it looks like a sort of Cauchy product

$$\sum_{q=0}^{+\infty}\beta_q \cdot\sum_{q=0}^{+\infty}\gamma_q=\sum_{q=0}^{+\infty}\sum_{r=0}^q\beta_r \gamma_{q-r}$$

but of course is missing the term $\alpha_q$ unfortunately. Do you have any idea?

Answer 1

The common representations are \begin{align*} \sum_{q=0}^{\infty}\alpha_q\sum_{r=0}^q\beta_r\gamma_{q-r} &=\sum_{\color{blue}{0\leq r\leq q<\infty}}\alpha_q\beta_r\gamma_{q-r}\\ &=\sum_{r=0}^\infty\beta_r\sum_{q=r}^\infty\alpha_q\gamma_{q-r}\\ &=\sum_{r=0}^\infty\beta_r\sum_{q=0}^\infty\alpha_{q+r}\gamma_{q}\\ \end{align*}