A Further Extension for Ramanujan’s Beta Integral and Applications
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 < q < 1 , x > 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.
2010 ◽
Vol 365
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pp. 653-658
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1982 ◽
Vol 85
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pp. 360
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2013 ◽
Vol 104
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pp. 397-414
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2008 ◽
Vol 281
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pp. 626-644
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