Let H denote the hypergeometric 3 F 2 function over F p whose three numerator parameters are quadratic characters and whose two denominator parameters are trivial characters. In 1992, Koike posed the problem of evaluating H at the argument 1/4. This problem was solved by Ono in 1998. Ten years later, Evans and Greene extended Ono's result by evaluating an… (More)
Key words and phrases. Hypergeometric functions over finite fields, elliptic curves with complex multiplication, orders in imaginary quadratic fields, ring class fields, genus class fields, Ramanujan class invariants. Abstract For an odd prime p, define H p (z) = u,v(mod p) uv(1−u)(1−v)(1−uvz) p , where z is an integer (mod p) and the summands are Legendre… (More)
We prove a general identity for a 3 F 2 hypergeometric function over a finite field F q , where q is a power of an odd prime. A special case of this identity was proved by Greene and Stanton in 1986. As an application, we prove a finite field analogue of Clausen's Theorem expressing a 3 F 2 as the square of a 2 F 1. As another application, we evaluate an… (More)
We generalize two mind-switching problems that arise in connection with the popular sci-fi television series Futurama and Stargate SG-1. Optimal solutions to these problems are found by answering the following question about a permutation σ expressed as a product of nontrivial disjoint cycles. " When writing σ as a product of distinct transpositions none… (More)
We prove two general formulas for a two-parameter family of hyperge-ometric 3 F 2 (z) functions over a finite field Fq, where q is a power of an odd prime. Each formula evaluates a 3 F 2 in terms of a 2 F 1 over F q 2. As applications, we evaluate infinite one-parameter families of 3 F 2 (1 4) and 3 F 2 (−1), thereby extending results of J. Greene–D.… (More)
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To Roger Howe on his sixtieth birthday. Abstract. The purpose of this paper is to develop optimal strategies for a simple integer choice game with a skew symmetric payoff matrix. The analysis involves the calculation of certain Pfaffians associated with these matrices.