A common recursive formula for Ramanujan’s ₁𝜓₁ and Bailey’s very-well-poised ₆𝜓₆ summation formulas

Abstract

In this paper, we establish a general recursive formula via the usual telescoping method for certain bilateral truncated sum which may serve as a common source for Ramanujan’s 1 ψ 1 {}_1\psi _1 , Bailey’s very-well-poised 6 ψ 6 {}_6\psi _6 , and Rogers’ 6 ϕ 5 {}_6\phi _5 summation formulas. The corresponding derivations for these summation formulas are presented in details.