A class of generalized hypergeometric summations

Summations over the positive integers n of the generalized hypergeometric expressions (±1)npFp+1[−n2x2] (x > 0) are derived in closed form. The specialization p = 0, for example, reduces to known results for Schlömilch series. In addition, we record the apparently not readily available sine and cosine transforms of pFp+1[−b2x2] (b > 0), the latter of which is used together with a form of the Poisson summation formula to deduce the aforementioned results.