Algebraicity conjecture for the auxiliary series of the seven Weyl models

Consider the seven Weyl models in Table~

, and let $A(x,y)$ be the series defined in Proposition~

. Weyl-model algebraicity conjecture. For every one of these seven models, $A(x,y)$ is algebraic; consequently, $C(x,y)$ is D-finite. In addition, for each of the four models in Table~

, $C(x,y)$ is algebraic, as is $Q(x,y)$. The conjecture is proved for three of the seven Weyl models and for three of the four models in Table~

; five models therefore remain open, including four Weyl models and Gessel's model.