Nonnegativity of q-hypergeometric series

Question

What are methods for proving nonnegativity of q-hypergeometric functions? Specifically, I have a function of the type 4-phi-3, it is a terminating series: $$ {}_{4}\phi_3\left(\begin{matrix} q^{-i_1},q^{-j_1},zs_1^{-1}s_2 ,q z^{-1}s_1^{-1}s_2\\ s_2^{2},q^{1+j_2-j_1},s_1^{-2}q^{1-i_1-j_2}\end{matrix} \bigg|\, q,q\right). $$ I know from numerics that it should be nonnegative if $0<q<1, -1<s_1,s_2<0$, and $0\le z\le \min(s_1/s_2,s_2/s_1)$. Here $i_1,i_2,j_1,j_2$ are nonnegative integers with condition $i_1+j_2=j_1+i_2$.

However, in the expansion of the function the individual terms are not all nonnegative. They seem to be of alternating signs, and are not monotone either. I tried various Watson's transformations turning the function into 8-phi-7, but so far did not succeed. Any help appreciated.

Answer 1

In this paper (https://arxiv.org/abs/1905.06815) there is a similar 4-phi-3 nonnegativity statement which in fact can be utilized to get the nonnegativity of the function in question. In Proposition A.8 we prove something very similar for an expression defined in (A.14), and there Watson's transformation works indeed.

I guess when posting the question I did not carefully check this first.