A Finite Analogue Of Fine's Function $F(a,b;t)$

We initiate a systematic development of $F_N(a,b;t)$, a finite analogue of Fine’s function $F(a,b;t)$. Our results are transformations between $F_N(a,b;t)$ and $F_N(a{q}^{\mathrm{\ell }},b{q}^m;t{q}^n)$, where $\mathrm{\ell },m$ and $n$ take the values $0$ or $1$.