On transformation formulas of $p$-adic hypergeometric functions $\mathscr{F}_{a,...,a}^{(\sigma)}(t) $ and $\widehat{\mathscr{F}}_{a,...,a}^{(\sigma)}(t)$

TL;DR

This paper explores transformation formulas of $p$-adic hypergeometric functions, establishing implications among different formulas and enhancing understanding of their structural relationships.

Contribution

It demonstrates that certain transformation formulas of $p$-adic hypergeometric functions imply others, clarifying their interconnectedness.

Findings

Proved implication relations among transformation formulas

Enhanced understanding of structural relationships of $p$-adic hypergeometric functions

Recalled key hypergeometric functions and their transformation properties

Abstract

In this paper, we recall hypergeometric functions $\mathscr{F}^{\rm Dw}_{a_1,\cdots,a_s}(t),$ $\mathscr{F}^{(\sigma)}_{a_1,\cdots,a_s}(t)$, $\widehat{\mathscr{F}}^{(\sigma)}_{a,\cdots,a}(t)$ and their transformation formulas. Then we prove that one of transformation formula implies another.