The three-variable basic hypergeometric conjecture for Macdonald eigenfunctions

Let $q,t,s_1,s_2,s_3$ be parameters, let $x_1,x_2,x_3$ be variables, and let $D^1$ be the difference operator defined earlier. Write $(a;q)_k$ for the $q$-shifted factorial and ${}_2\phi_1$ for the basic hypergeometric series

{}_2\phi_1\left(\begin{matrix}a,b\c\end{matrix};q,z\right)=\sum_{n=0}^{\infty}\frac{(a;q)_n(b;q)_n}{(q;q)_n(c;q)_n}z^n.

Three-variable basic hypergeometric conjecture. The series

$$\begin{aligned} f(x_1,x_2,x_3;&s_1,s_2,s_3,q,t)\\ ={}&\sum_{k=0}^{\infty}\frac{(qt^{-1};q)_k^2(t;q)_k^2}{(q;q)_k(qs_1/s_2;q)_k(qs_2/s_3;q)_k(qs_1/s_3;q)_k}(qs_1/s_3)^k(x_3/x_1)^k\\ &\times\prod_{1\leq i<j\leq3}(1-x_j/x_i) \,{}_2\phi_1\left(\begin{matrix}q^{k+1}t^{-1},qt^{-1}s_i/s_j\q^{k+1}s_i/s_j\end{matrix};q,t x_j/x_i\right) \end{aligned}$$

satisfies the difference equation for $n=3$. This is a special three-variable formula for the conjectural eigenfunction series; its relation to the general coefficient expansion was not proved in the paper, although agreement was checked to certain degrees.