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We principally present reductions of certain generalized hypergeometricfunctions $_3F_2(\pm 1)$ in terms of products of elementary functions. Most ofthese results have been known for some time, but one of the methods, wherein wesimultaneously solve for three alternating binomial sums, may be new. We obtaina functional equation holding for all three of this set of alternating binomialsums. Using successive derivatives, we show how related chains of $_3F_2(\pm1)$ values may be obtained. It may be emphasized that we make no reliance onthe WZ method for hypergeometric summation. Additional material on Pochhammersymbols and certain of their products is presented in an Appendix to supplementthe pedagogical content of the paper.

Reductions of particular hypergeometric functions $_3F_2(a,a+1/3,a+2/3;p/3,q/3;\pm 1)$