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In 1915, Ramanujan stated the following formula ∫ 0 ∞ t x − 1 ( − a t ; q ) ∞ ( − t ; q ) ∞ d t = π sin π x ( q 1 − x , a ; q ) ∞ ( q , a q − x ; q ) ∞ , where 0 0 , and 0 < a < q x . The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As some applications, we obtain some new integral formulas of Ramanujan and also show some new representation with gamma functions and q-gamma functions.

A Further Extension for Ramanujan’s Beta Integral and Applications