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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.

    diff, v8, current

    • 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

    diff, v12, current

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeOct 16th 2023
    • (edited Oct 16th 2023)

    Hypergeometric function F= 2F 1(a,b;c;x)F = {}_2 F_1(a,b;c;x) satisfies the differential equation

    x(1x)d 2Fdx 2+[c(a+b1)x]dFdxabF=0. 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)>0Re(c)\gt 0, Re(b)>0Re(b)\gt 0 function 2F 1(a,b;c;x){}_2 F_1(a,b;c;x) can be represented as the Euler integral

    2F 1(a,b;c;x)=Γ(c)Γ(b)Γ(cb) 0 1t b(1t) cb1(1tx) adt,x[1,+). {}_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 11. The second solution of the differential equation around 00 is x 1c 2F 1(ac+1,bc+1,2c;x)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 aF(a,1c+1,1b+a,x 1)x^{-a}F(a,1-c+1,1-b+a, x^{-1}) and x bF(b,1c+b,1a+b;x 1).x^{-b}F(b,1-c+b,1-a+b;x^{-1})..

    diff, v13, current

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

    x 1c 2F 1(ac+1,bc+1;2c;x)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.