SOME BASIC BILATERAL SUMS AND INTEGRALS

By splitting the real line into intervals of unit length a doubly infi nite integral of the form integral(-infinity)(infinity) F(q(x)) dx, 0 < q < 1, can clearly be expressed as integral(0)(1) Sigma(n=1 infinity )(infinity) = F(q(x+n)) dx, provided F satisfies the appropriate condi tions. This simple idea is used to prove Ramanujan's integral analogue s of his 1 psi 1 sum and give a new proof of Askey and Boy's extention of it. Integral analogues of the well-poised 2 psi 2 sum as well as t he very-well-poised 6 psi 6 sum are also found in a straightforward ma nner. An extension to a very-well-poised and balanced 8 psi 8 series i s also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.