A symmetric formula for hypergeometric series

In terms of Dougall’s 2 H 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 Saalschütz’s theorem. Similarly, we also show that Bailey’s 6 ψ 6 series identity implies the nonterminating form of Jackson’s 8 ϕ 7 summation formula. Considering the reversibility of the proofs, it is routine to show that Dougall’s 2 H 2 series identity is equivalent to a known nonterminating form of Saalschütz’s theorem and Bailey’s 6 ψ 6 series identity is equivalent to the nonterminating form of Jackson’s 8 ϕ 7 summation formula.