I thought about continued fractions as a cool way to represent numbers, but also about the fact they are difficult to treat because standard algebraic operations like addition and multiplication don't work on them in a simple way. My question is: do there exist some simple and interesting operations that have a regular behavior with respect to the normal form of a continued fraction? For example, does there exist a simple $\circ$, $?$ or $*$ that satisfies
$$ a = [a_0; a_1, a_2 \, ... \, a_n] \\ b = [b_0; b_1, b_2 \, \ldots \, b_n] \\ a \circ b = [a_0 \circ b_0; a_1 \circ b_1, a_2 \circ b_2 \, \ldots \, a_n \circ b_n] $$ or $$ a \,?\, b = [a_0 + b_0; a_1 + b_1, a_2 + b_2 \, \ldots \, a_n + b_n] $$ or $$ *a = [2 a_0; 2 a_1, 2 a_2 \, \ldots \, 2 a_n] $$
Note that $ [a_0; a_1, a_2 \, \ldots \, a_n] $ can represent:
$$ a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{\ddots a_{n-1} + \cfrac{1}{a_n}}}} $$ or $$ a_0 + \cfrac{m_1}{a_1 + \cfrac{m_2}{a_2 + \cfrac{m_3}{\ddots a_{n-1} + \cfrac{m_{n}}{a_n}}}} $$