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

TL;DR

This paper introduces a finite analogue of Fine's function, developing transformations between these finite versions and exploring their properties.

Contribution

It systematically develops the finite analogue $F_N(a, b; t)$ of Fine's function and establishes key transformations between related finite functions.

Findings

Derived transformations between $F_N(a, b; t)$ and shifted versions

Established properties of the finite analogue

Paved the way for further finite analogue studies

Abstract

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 $F_N(aq^{\ell}, bq^{m}; tq^{n})$, where $\ell,m$ and $n$ take the values $0$ or $1$.