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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMay 4th 2020

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.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeNov 3rd 2022
• 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
• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeOct 16th 2023
• (edited Oct 16th 2023)

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 $x^{-b}F(b,1-c+b,1-a+b;x^{-1}).$.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeOct 16th 2023
• (edited Oct 16th 2023)

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

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeOct 16th 2023

I also done some reformatting, addressing partially concerns in 1.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeOct 16th 2023

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