Non-D-finiteness conjecture for Models 1, 3, 4, 17, 18 and 22

Let $G(t;a,b)$ denote the generating function for the corresponding model, with parameters $t$, $a$, and $b$. A formal power series is D-finite if it satisfies a linear differential equation with polynomial coefficients. Non-D-finiteness conjecture. The generating functions $G(t;a,a)$ and $G(t;a,b)$ for Models 1, 3, 4, 17, 18 and 22 are not D-finite.

The conjecture is based on closed-form expressions involving ratios or reciprocals of D-finite functions, which suggest D-algebraic but non-D-finite solutions. No proof of non-D-finiteness or suitable D-finite solutions has been obtained for these models.