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Examples

(sum_{1 <= i <= n} xⁱ) * (sum_{1 <= j <= m} x^j) = (sum_{1 <= i <= n} (sum_{1 <= j <= m} x^i+j)) = (sum_{1 <= i <= n /\  1 <= j <= m} x^i+j).

(sum_{1 <= i <= n} (sum_{1 <= j <= m} i*x^j)) = (sum_{1 <= i <= n} (i*(sum_{1 <= j <= m} x^j))).

Many more identities can be deduced from basic definition.