Got these problems listed in a text book. I can't figure out what the meaning of the sigma notation is here. How did they obtain those expansions of $\sum\alpha^3\beta$ & $\sum \frac{\alpha}{\beta}$? I know that sigma means summation, but usually it contains some variables and indices. Can anyone decipher this.
Example Problems
Q1. If $\alpha, \beta, \gamma$ be the roots of the equation $x^3-px^2+qx-\gamma=0$. Find $\sum\alpha^3\beta$.
A1. $$\sum\alpha^3\beta=\alpha^3(\beta+\gamma)+\alpha^3(\gamma+\alpha)+\alpha^3(\alpha+\beta)$$ $$\sum\alpha^3\beta=(\alpha+\beta+\gamma)(\beta\gamma+\gamma\alpha+\alpha\beta)-3\alpha\beta\gamma$$ $$\sum\alpha^3\beta=pq-3\gamma$$
Q2. If $\alpha, \beta, \gamma$ be the roots of the equation $x^3-px^2+9x-r=0$ where $r\ne0$. Find $\sum \frac{\alpha}{\beta}$.
A2. $$\sum \frac{\alpha}{\beta}=\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\alpha}{\gamma}+\frac{\gamma}{\alpha}+\frac{\beta}{\gamma}+\frac{\gamma}{\beta}$$ $$...$$ $$=\frac{pq}{r}-3$$
That is quite strange. Normally, these codensed notaions proceed by circular permutation of the roots, i.e. \begin{align} \sum \alpha^3\beta &= \alpha^3\beta +\beta^3\gamma + \gamma^3\alpha,\\ \sum \frac\alpha\beta &= \frac\alpha\beta + \frac\beta\gamma + \frac\gamma \alpha=\frac{\alpha\beta^2+\beta\gamma^2+\gamma\alpha^2}{\alpha\beta\gamma }. \end{align}