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h21-0041

A relation at x=1 of Kummer's 24 solutions

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

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

A relation at x=1 of Kummer's 24 solutions

Reference: [3, 38-39]

Reference: [4, 64-65] Retrieve the formula in Mathematica form h21-0041-math-auto.m Retrieve the formula in Risa/Asir form h21-0041-asir-auto.rr Retrieve the formula in LaTeX form h21-0041-tex-auto.tex Interactive replacement h21-0041-js-auto.html


Nobuki Takayama 2003-02-03