ACTA ARITHMETICA LXVII.2 (1994)

On special values of generalized p-adic hypergeometric functions by Kaori Ota (Tokyo) We generalize results for ratios of generalized hypergeometric functions obtained by P. T. Young [9] to the case of a power of p. More precisely, we generalize Theorems 3.2 and 3.4 in [9]. In order to do that, introducing a generalization of the p-adic Gamma function to a power of p seems to simplify expressions of the results, and this is done in Section 1. There we prove properties of this generalized Gamma function. In Section 2, after proving some preliminary results for certain ratios of binomial coefficients and giving a “truncated ” version of Dwork’s theorem, we prove the theorems. We follow very closely the proofs of the corresponding theorems in [9], and our methods also apply to the well-poised series (Theorems 3.1 and 3.3 in [9]) treated in [10]. Some applications are also given in [10]. Throughout this paper, p is a prime, νp denotes the p-adic valuation normalized by νp(p) = 1 and by the p-adic expansion we mean the standard p-adic expansion whose coefficients are all in {0, 1,..., p − 1}. 1. The p-adic Gamma function is defined by Γp(n+ 1) = (−1)n+1 n∏ j=1 p- j j for a positive integer n and is extended by continuity to all of Zp: Γp: Zp → Z∗p (cf. [6]). Here Zp is the ring of p-adic integers. For a non-negative integer l, we define a map hl from Zp to Zp by hl(x) = j≥l xjp j−l for x ∈ Zp with the p-adic expansion j=0 xjp j. Then hl is obviously con-tinuous. Let t be a positive integer and q = pt. [141] 142 K. Ota Definition. For x ∈ Zp, we define a map Γq from Zp to Z∗p by Γq(x+ 1) = t−1∏ l=0 Γp(hl(x) + 1). Then Γq is continuous and, for t = 1, Γq coincides with Γp. N. Koblitz con-structed and studied an extension of the p-adic Gamma function (cf. [4], [5]). There he uses Γq for a different function, but this will cause no confusion. Γq has the following properties analogous to Γp. Proposition 1. (1) If x ≡ y (mod pn) for n ≥ t − 1, then Γq(x) ≡ Γq(y) (mod pn−(t−1)). (2) For a positive integer n, Γq(n+ 1) = (−1)A p−B n∏ j=1 q- j j, where A = t+ t−1∑ i=