Values of \(p\)-adic hypergeometric functions and \(p\)-adic analogue of Kummer's linear identity (Q6628932)

The author studies McCarthy's hypergeometric sums \(p\)-adically and expresses them in terms of the roots of an explicit polynomial over \(\mathbb{F}_p\). Several corollaries follow.\N\NLet \(p\) be an odd prime, \(n\geq 1\) and \(t\in\mathbb{F}_p\). The family of hypergeometric sums considered by the author is:\N\[\N_{3n-1}G_{3n-1}(p,t) = _{3n-1} \mathbb{G}_{3n-1}\left [ \begin{matrix} \frac{1}{3n}, & \frac{2}{3n}, & \frac{3}{3n}, & \cdots, & \frac{3n-1}{3n}\\\N0, & \frac{1}{2}, & \frac{1}{3n-2}, & \cdots, & \frac{3n-3}{3n-2}\end{matrix} \biggr\rvert t\right ]_p.\N\]\NLet \(\widehat{\mathbb{F}_p^\times}\) be the group of multiplicative characters of \(\mathbb{F}_p^\times\) and define a map \(\delta\) from \(\widehat{\mathbb{F}_p^\times}\to\{0,1\}\) by setting \(\delta(\chi)=1\) if \(\chi\) is the trivial character and zero otherwise. The main theorem states that if \(p\nmid 3n(3n-2)\), \(\alpha = \frac{(-1)^n(3n-2)^{3n-2}}{(3n)^{3n}t}\) and\N\[\Nf_t(y)=y^{3n}-2y^{3n-1}+y^{3n-2}-(-1)^n4\alpha\in\mathbb{F}_p[y],\N\]\Nthen:\N\[\N_{3n-1}G_{3n-1}(p,t) = r-1 + \frac{1-p}{p}\phi(\alpha)\delta(\phi^n),\N\]\Nwhere \(\phi\) is the quadratic character and \(r\) is the number of distinct roots of \(f_t(y)\) over \(\mathbb{F}_p\).\N\NAs an example of an immediate corollary, when \(n=1\), it is shown that \(_2G_2(p,1)=1\), and when \(t\neq 1\), \(_2G_2(p,t)\) is equal to one less the cardinality in \(\{y\in\mathbb{F}_p : 27y^3-27y^2+4t^{-1}\equiv 0\bmod p\}\).\N\NAnalogous results are shown for the function\N\[\N_2\widetilde{G}_2(p,t) = _2\mathbb{G}_2\left [\begin{matrix} \frac{1}{6}, & \frac{5}{6}\\\N0, &\frac{1}{2}\end{matrix} \biggr\rvert t\right]_p,\N\]\Nand it is also shown that for \(p>3\), \(_2\widetilde{G}_2(p,t)\) can only have values \(0,\pm 1\) and \(\pm 2\).\N\NFinally, a \(p\)-adic version of Kummer's transformation identity is given between \(_2G_2\) and \(_2\widetilde{G}_2\).