Some basic bilateral sums and integrals

Abstract

By splitting the real line into intervals of unit length a doubly infinite integral of the form $\Int F(q^x)\,dx,\; 0<q<1$, can clearly be expressed as $\Integ \Sum F(q^{x+n})\,dx$, provided $F$ satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his \ph{1}{1} sum and give a new proof of Askey and Roy's extention of it. Integral analogues of the well-poised \ph{2}{2} sum as well as the very-well-poised \ph{6}{6} sum are also found in a straightforward manner. An extension to a very-well-poised and balanced \ph{8}{8} series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given.