A Further Extension for Ramanujan’s Beta Integral and Applications

oleh: Gao-Wen Xi, Qiu-Ming Luo

Format: Article
Diterbitkan: MDPI AG 2019-01-01

Deskripsi

In 1915, Ramanujan stated the following formula <inline-formula> <math display="inline"> <semantics> <mrow> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>∞</mi> </msubsup> <msup> <mi>t</mi> <mrow> <mi>x</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <msub> <mrow> <mo>(</mo> <mo>-</mo> <mi>a</mi> <mi>t</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo> </mrow> <mi>∞</mi> </msub> <msub> <mrow> <mo>(</mo> <mo>-</mo> <mi>t</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mrow> <mo form="prefix">sin</mo> <mi>π</mi> <mi>x</mi> </mrow> </mfrac> <mfrac> <msub> <mrow> <mo>(</mo> <msup> <mi>q</mi> <mrow> <mn>1</mn> <mo>-</mo> <mi>x</mi> </mrow> </msup> <mo>,</mo> <mi>a</mi> <mo>;</mo> <mi>q</mi> <mo>)</mo> </mrow> <mi>∞</mi> </msub> <msub> <mrow> <mo>(</mo> <mi>q</mi> <mo>,</mo> <mi>a</mi> <msup> <mi>q</mi> <mrow> <mo>-</mo> <mi>x</mi> </mrow> </msup> <mo>;</mo> <mi>q</mi> <mo>)</mo> </mrow> <mi>∞</mi> </msub> </mfrac> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>q</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>a</mi> <mo>&lt;</mo> <msup> <mi>q</mi> <mi>x</mi> </msup> </mrow> </semantics> </math> </inline-formula>. The above formula is called Ramanujan’s beta integral. In this paper, by using <i>q</i>-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 <i>q</i>-gamma functions.