By splitting the real line into intervals of unit length a doubly infinite integral of the form J (^>oo F{q x) dx, 0 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)
Ismail et al. (Sun,) studied this question.