Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers
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></semanti...
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MDPI AG
2022-10-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3903 |
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| author | Alexander Dyachenko Dmitrii Karp |
| author_facet | Alexander Dyachenko Dmitrii Karp |
| author_sort | Alexander Dyachenko |
| collection | DOAJ |
| description | 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. |
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| language | English |
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| publishDate | 2022-10-01 |
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| spelling | doaj.art-35db748bc3dd46ba8da884253a8a56d82023-11-24T01:08:17ZengMDPI AGMathematics2227-73902022-10-011020390310.3390/math10203903Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by IntegersAlexander Dyachenko0Dmitrii Karp1Keldysh Institute of Applied Mathematics, 125047 Moscow, RussiaDepartment of Mathematics, Holon Institute of Technology, Holon 5810201, IsraelGiven 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.https://www.mdpi.com/2227-7390/10/20/3903gauss hypergeometric functiongauss continued fractionintegral representation |
| spellingShingle | Alexander Dyachenko Dmitrii Karp Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers Mathematics gauss hypergeometric function gauss continued fraction integral representation |
| title | Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers |
| title_full | Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers |
| title_fullStr | Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers |
| title_full_unstemmed | Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers |
| title_short | Integral Representations of Ratios of the Gauss Hypergeometric Functions with Parameters Shifted by Integers |
| title_sort | integral representations of ratios of the gauss hypergeometric functions with parameters shifted by integers |
| topic | gauss hypergeometric function gauss continued fraction integral representation |
| url | https://www.mdpi.com/2227-7390/10/20/3903 |
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