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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory object of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • 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

2. • 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

3. • 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 $F = {}_2 F_1(a,b;c;x)$ 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 $x^{-b}F(b,1-c+b,1-a+b;x^{-1}).$.

   diff, v13, current

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

   $x^{1-c} {}_2 F_1(a-c+1,b-c+1;2-c;x)$

5. • 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.

6. • 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.