| Summary: | Given real parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></mrow></semantics></math></inline-formula> and integer shifts <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>n</mi><mn>1</mn></msub><mo>,</mo><msub><mi>n</mi><mn>2</mn></msub><mo>,</mo><mi>m</mi></mrow></semantics></math></inline-formula>, we consider the ratio <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>=</mo><msub><mrow></mrow><mn>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>a</mi><mo>+</mo><msub><mi>n</mi><mn>1</mn></msub><mo>,</mo><mi>b</mi><mo>+</mo><msub><mi>n</mi><mn>2</mn></msub><mo>;</mo><mi>c</mi><mo>+</mo><mi>m</mi><mo>;</mo><mi>z</mi><mo>)</mo></mrow><mo>/</mo><msub><mrow></mrow><mn>2</mn></msub><msub><mi>F</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>z</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> of the Gauss hypergeometric functions. We find a formula for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Im</mi><mi>R</mi><mo>(</mo><mi>x</mi><mo>±</mo><mi>i</mi><mn>0</mn><mo>)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> in terms of real hypergeometric polynomial <i>P</i>, beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for <i>R</i> when the asymptotic behaviour at unity is mild and the denominator does not vanish. The results are illustrated with a large number of examples.
|
|---|