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nLab > Latest Changes: hypergeometric function

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- CommentRowNumber1.
- CommentAuthorUrs
- CommentTimeMay 4th 2020
- PermaLink
Author: Urs
Format: MarkdownItexhave touched this old entry for formatting, hyperlinking, grammar and spelling. The two comment paragraphs below the definition should probably either go up into the Idea-section and else be moved to their own Outloook-section or similar. Not sure. <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+function/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+function/8">v8</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+function">current</a>

have touched this old entry for formatting, hyperlinking, grammar and spelling. The two comment paragraphs below the definition should probably either go up into the Idea-section and else be moved to their own Outloook-section or similar. Not sure.

diff, v8, current
- CommentRowNumber2.
- CommentAuthorzskoda
- CommentTimeNov 3rd 2022
- PermaLink
Author: zskoda
Format: MarkdownItex* [[Israel Gelfand|I. M. Gel’fand]], M. I. Graev, [[Vladimir Retakh|V. S. Retakh]], _General hypergeometric systems of equations and series of hypergeometric type_, Russian Math. Surveys __47__(4) (1992) 1--88 [doi](https://doi.org/10.1070/RM1992v047n04ABEH000915), transl. from _Общие гипергеометрические системы уравнений и ряды гипергеометрического типа_, УМН __47__:4 (1992) 3--82; _General gamma functions, exponentials, and hypergeometric functions_, Russian Math. Surveys __53__:1 (1998) 1--55 [doi](https://doi.org/10.1070/RM1998v053n01ABEH000008), transl. from _Общие гамма-функции, экспоненты и гипергеометрические функции_, УМН, 1998, __53__:1 (319) 3--60 [doi](https://doi.org/10.4213/rm8) Успехи математических наук __53__:1 (319) (1998) 3--60 <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+function/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+function/12">v12</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+function">current</a>
- I. M. Gel’fand, M. I. Graev, V. S. Retakh, General hypergeometric systems of equations and series of hypergeometric type, Russian Math. Surveys 47(4) (1992) 1–88 doi, transl. from Общие гипергеометрические системы уравнений и ряды гипергеометрического типа, УМН 47:4 (1992) 3–82; General gamma functions, exponentials, and hypergeometric functions, Russian Math. Surveys 53:1 (1998) 1–55 doi, transl. from Общие гамма-функции, экспоненты и гипергеометрические функции, УМН, 1998, 53:1 (319) 3–60 doi Успехи математических наук 53:1 (319) (1998) 3–60
diff, v12, current
- CommentRowNumber3.
- CommentAuthorzskoda
- CommentTimeOct 16th 2023
- (edited Oct 16th 2023)
- PermaLink
Author: zskoda
Format: MarkdownItexHypergeometric function $F = {}_2 F_1(a,b;c;x)$ satisfies the differential equation $$ x(1-x)\frac{d^2 F}{d x^2} + [c - (a+b-1)x]\frac{d F}{d x} - a b F = 0. $$ For $Re(c)\gt 0$, $Re(b)\gt 0$ function ${}_2 F_1(a,b;c;x)$ can be represented as the Euler integral $$ {}_2 F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^b (1-t)^{c-b-1}(1-t x)^{-a} d t,\,\,\,\,x\notin [1,+\infty). $$ The value of this function at origin is $1$. The second solution of the differential equation around $0$ is $x^{1-c} {}_2 F_1(a-c+1,b-c+1,2-c;x)$. The basis of solutions around $\infty$ is given by $x^{-a}F(a,1-c+1,1-b+a, x^{-1})$ and $x^{-b}F(b,1-c+b,1-a+b;x^{-1}).$. <a href="https://ncatlab.org/nlab/revision/diff/hypergeometric+function/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/hypergeometric+function/13">v13</a>, <a href="https://ncatlab.org/nlab/show/hypergeometric+function">current</a>

Hypergeometric function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>F</mi><mo>=</mo><msub><mrow/> <mn>2</mn></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F = {}_2 F_1(a,b;c;x)</annotation></semantics></math>$ satisfies the differential equation
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>x</mi><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup><mi>F</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mn>2</mn></msup></mrow></mfrac><mo>+</mo><mo stretchy="false">[</mo><mi>c</mi><mo>−</mo><mo stretchy="false">(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mi>x</mi><mo stretchy="false">]</mo><mfrac><mrow><mi>d</mi><mi>F</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>−</mo><mi>a</mi><mi>b</mi><mi>F</mi><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> x(1-x)\frac{d^2 F}{d x^2} + [c - (a+b-1)x]\frac{d F}{d x} - a b F = 0. </annotation></semantics></math>$
For $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Re</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Re(c)\gt 0</annotation></semantics></math>$ , $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Re</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>&gt;</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Re(b)\gt 0</annotation></semantics></math>$ function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow/> <mn>2</mn></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{}_2 F_1(a,b;c;x)</annotation></semantics></math>$ can be represented as the Euler integral
$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mrow/> <mn>2</mn></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>−</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></mfrac><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><msup><mi>t</mi> <mi>b</mi></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><msup><mo stretchy="false">)</mo> <mrow><mi>c</mi><mo>−</mo><mi>b</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mi>x</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>a</mi></mrow></msup><mi>d</mi><mi>t</mi><mo>,</mo><mspace width="0.16667em"/><mspace width="0.16667em"/><mspace width="0.16667em"/><mspace width="0.16667em"/><mi>x</mi><mo>∉</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mo lspace="0.11111em" rspace="0em">+</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> {}_2 F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}\int_0^1 t^b (1-t)^{c-b-1}(1-t x)^{-a} d t,\,\,\,\,x\notin [1,+\infty). </annotation></semantics></math>$
The value of this function at origin is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>$ . The second solution of the differential equation around $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>$ is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi> <mrow><mn>1</mn><mo>−</mo><mi>c</mi></mrow></msup><msub><mrow/> <mn>2</mn></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>−</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^{1-c} {}_2 F_1(a-c+1,b-c+1,2-c;x)</annotation></semantics></math>$ . The basis of solutions around $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>$ is given by $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>a</mi></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>−</mo><mi>b</mi><mo>+</mo><mi>a</mi><mo>,</mo><msup><mi>x</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^{-a}F(a,1-c+1,1-b+a, x^{-1})</annotation></semantics></math>$ and $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mi>b</mi></mrow></msup><mi>F</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>c</mi><mo>+</mo><mi>b</mi><mo>,</mo><mn>1</mn><mo>−</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>;</mo><msup><mi>x</mi> <mrow><mo lspace="0.11111em" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex">x^{-b}F(b,1-c+b,1-a+b;x^{-1}).</annotation></semantics></math>$ .

diff, v13, current
- CommentRowNumber4.
- CommentAuthorzskoda
- CommentTimeOct 16th 2023
- (edited Oct 16th 2023)
- PermaLink
Author: zskoda
Format: MarkdownItex $x^{1-c} {}_2 F_1(a-c+1,b-c+1;2-c;x)$

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>x</mi> <mrow><mn>1</mn><mo>−</mo><mi>c</mi></mrow></msup><msub><mrow/> <mn>2</mn></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>a</mi><mo>−</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo>+</mo><mn>1</mn><mo>;</mo><mn>2</mn><mo>−</mo><mi>c</mi><mo>;</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x^{1-c} {}_2 F_1(a-c+1,b-c+1;2-c;x)</annotation></semantics></math>$
- CommentRowNumber5.
- CommentAuthorzskoda
- CommentTimeOct 16th 2023
- PermaLink
Author: zskoda
Format: MarkdownItexI also done some reformatting, addressing partially concerns in 1.

I also done some reformatting, addressing partially concerns in 1.
- CommentRowNumber6.
- CommentAuthorzskoda
- CommentTimeOct 16th 2023
- PermaLink
Author: zskoda
Format: MarkdownItexI have improved hyperlinking in the entry. The entry looks better and more complete now.

I have improved hyperlinking in the entry. The entry looks better and more complete now.

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nLab > Latest Changes: hypergeometric function