Zeros of hypergeometric functions in the p-adic setting

Abstract

McCarthy (Pac J Math 261(1):219–236, 2013) defined hypergeometric functions in the p-adic setting over finite fields using p-adic gamma functions. These functions possess many properties that are analogous to classical hypergeometric type identities. In this paper, we investigate values of two generic families of these hypergeometric functions that we denote by $${_nG_n}(t)_p$$ n G n ( t ) p and $${_n{\widetilde{G}}_n}(t)_p$$ n G ~ n ( t ) p for $$n\ge 3$$ n ≥ 3 , and $$t\in {\mathbb {F}}_p$$ t ∈ F p , the finite field with p elements. These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We also examine zeros of the functions $${_nG_n}(t)_p$$ n G n ( t ) p , and $${_n{\widetilde{G}}_n}(t)_p$$ n G ~ n ( t ) p over $${\mathbb {F}}_p$$ F p . Moreover, we classify the values of t for which $${_nG_n}(t)_p=0$$ n G n ( t ) p = 0 for infinitely many primes. For example, we show that there are infinitely many primes for which $${_{2k}G_{2k}}(-1)_p=0$$ 2 k G 2 k ( - 1 ) p = 0 . In contrast, for $$t\ne 0$$ t ≠ 0 there are no primes for which $${_{2k}{\widetilde{G}}_{2k}}(t)_p=0$$ 2 k G ~ 2 k ( t ) p = 0 .