A symmetric formula for hypergeometric series

Abstract

In terms of Dougall’s $$_2H_2$$ series identity and the series rearrangement method, we establish a symmetric formula for hypergeometric series. Then it is utilized to derive a known nonterminating form of Saalschutz’s theorem. Similarly, we also show that Bailey’s $$_6\psi _6$$ series identity implies the nonterminating form of Jackson’s $$_8\phi _7$$ summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall’s $$_2H_2$$ series identity is equivalent to a known nonterminating form of Saalschutz’s theorem and Bailey’s $$_6\psi _6$$ series identity is equivalent to the nonterminating form of Jackson’s $$_8\phi _7$$ summation formula.