Some basic bilateral sums and integrals

By splitting the real line into intervals of unit length a doubly infinite integral of the form J ( ^>oo F{q x )dx, 0 < q < 1, can clearly be expressed as J o --0O P (Q x * n )dx, provided F satisfies the appropriate conditions. This simple idea is used to prove Ramanujan's integral analogues of his sum and give a new proof of Askey and Roy's extent ion of it. Integral analogues of the well-poised 2^2 sum as well as the very-wellpoised 6^6 sum are also found in a straightforward manner. An extension to a very-well-poised and balanced 8^8 series is also given. A direct proof of a recent q-beta integral of Ismail and Masson is given. 1. Introduction. The familiar form of the classical beta integral of Euler is Jo ( + 0) Re (, 6) > 0. A less familiar form, obtained by a simple change of variable, is t a-i d t (1.2) B (a,b)= f-Jo U" U+6 ' There have been many extensions of both these forms, see, for example, Askey [2-5], Askey and Roy [6], Gasper [9, 10], Rahman and Suslov [18], and the references therein. A "curious" extension of (1.2) that was given by Ramanujan [21] in 1915 is Jo H; 9 )