The satellite formula for GM series

Let $K$ be a knot and $P$ a pattern whose pattern-defining link $P'$ has a well-defined GM series. Denote the satellite by $C_P(K)$. Satellite Formula. The GM series of $C_P(K)$ is given by either the formula in equation

$$or that in equation$$

, provided at least one converges; if both converge, the two expressions are equal. This extends the plumbed-knot and plumbed-pattern calculation to arbitrary knots and patterns with well-defined GM series; convergence is the stated limitation.