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h21-0034

A relation at x=0 of Kummer's 24 solutions

$ ((x)^{( 1 -c)}\cdot  {}_2F_1(((a-c)+ 1 ),((b-c)+ 1 ),( 2 -c),x))=(((x)^{( 1 -...
...))^{(c-a- 1 )})\cdot  {}_2F_1(( 1 -b),((a+ 1 )-c),( 2 -c),\frac{x}{(x- 1 )})) $
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$ {x^{1 - c}} {}_2 F_1 (1 + a - c,1 + b - c,2 - c,x) = {{\left( 1 - x \right) }^{-1 - a + c}} {x^{1 - c}} {}_2 F_1 (1 - b,1 + a - c,2 - c,{\frac{x}{-1 + x}}) $
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Formula in the tfb format:

    x ~arith1.power~ (1 ~arith1.minus~ c) ~arith1.times~
    hypergeo1.hypergeometric2F1(a ~arith1.minus~ c ~arith1.plus~ 1,
      b ~arith1.minus~ c ~arith1.plus~ 1, 2 ~arith1.minus~ c,x) ~relation1.eq~
    ( x ~arith1.power~ (1 ~arith1.minus~ c) ~arith1.times~
      ((1 ~arith1.minus~ x) ~arith1.power~ (c ~arith1.minus~ a ~arith1.minus~ 
1)) ~arith1.times~
      hypergeo1.hypergeometric2F1(1 ~arith1.minus~ b,
        a ~arith1.plus~ 1 ~arith1.minus~ c, 2 ~arith1.minus~ c,
            x ~arith1.divide~ (x ~arith1.minus~ 1)));

A relation at x=0 of Kummer's 24 solutions

Reference: [3, 38-39]

Reference: [4, 64-65] Retrieve the formula in Mathematica form h21-0034-math-auto.m Retrieve the formula in Risa/Asir form h21-0034-asir-auto.rr Retrieve the formula in LaTeX form h21-0034-tex-auto.tex Interactive replacement h21-0034-js-auto.html


Nobuki Takayama 2003-02-03