Defining parameters of a hypergeometric motive (awaiting review)

A hypergeometric motive $H(A,B,t)$ in $M(\Q,\Q)$ is described by a hypergeometric family $H(A,B)$ in $M(\Q(t),t)$ along with a specialization point $t \in \Q^\times$. Here, $A$ and $B$ are disjoint multisets of positive integers and are the defining parameters of the hypergeometric family; they are subject to a constraint that $$ \sum_{a \in A} \phi(a) = \sum_{b \in B} \phi(b); $$ this common value is the degree of the family. The value $t$ along with $A$ and $B$ are the defining parameters of the hypergeometric motive $H(A,B,t)$.

The list $\alpha$ is obtained from $A$ by replacing each $a \in A$ with the list of all elements of $\Q \cap (0,1]$ with denominator $a$, then sorting into ascending order; the list $\beta$ is obtained from $B$ by the same procedure. Both $\alpha$ and $\beta$ have length equal to the degree.

The list $\gamma$ of nonzero integers in ascending order is characterized by the property $$ \frac{\prod_{g \in \gamma, g>0} (x^g-1)}{\prod_{g \in \gamma,g<0} (x^{-g}-1)} = \frac{\prod_{a \in \alpha} (x - e^{2 \pi i a})}{\prod_{b \in \beta} (x - e^{2\pi i b})} = \frac{\prod_{a \in A} \Phi_a(x)}{\prod_{b \in B} \Phi_b(x)} $$ where $\Phi_n(x)$ denotes the $n$-th cyclotomic polynomial.