DLMF:16.12.E3 (Q5248)

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DLMF:16.12.E3
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    Defining formula
    ( F 1 2 ⁡ ( a , b c ; z ) ) 2 = ∑ k = 0 ∞ ( 2 ⁢ a ) k ⁢ ( 2 ⁢ b ) k ⁢ ( c - 1 2 ) k ( c ) k ⁢ ( 2 ⁢ c - 1 ) k ⁢ k ! ⁢ F 3 4 ⁡ ( - 1 2 ⁢ k , 1 2 ⁢ ( 1 - k ) , a + b - c + 1 2 , 1 2 a + 1 2 , b + 1 2 , 3 2 - k - c ; 1 ) ⁢ z k , superscript Gauss-hypergeometric-F-as-2F1 𝑎 𝑏 𝑐 𝑧 2 superscript subscript 𝑘 0 Pochhammer 2 𝑎 𝑘 Pochhammer 2 𝑏 𝑘 Pochhammer 𝑐 1 2 𝑘 Pochhammer 𝑐 𝑘 Pochhammer 2 𝑐 1 𝑘 𝑘 Gauss-hypergeometric-pFq 4 3 1 2 𝑘 1 2 1 𝑘 𝑎 𝑏 𝑐 1 2 1 2 𝑎 1 2 𝑏 1 2 3 2 𝑘 𝑐 1 superscript 𝑧 𝑘 {\displaystyle{\displaystyle\left({{}_{2}F_{1}}\left({a,b\atop c};z\right)% \right)^{2}=\sum_{k=0}^{\infty}\frac{{\left(2a\right)_{k}}{\left(2b\right)_{k}% }{\left(c-\frac{1}{2}\right)_{k}}}{{\left(c\right)_{k}}{\left(2c-1\right)_{k}}% k!}{{}_{4}F_{3}}\left({-\frac{1}{2}k,\frac{1}{2}(1-k),a+b-c+\frac{1}{2},\frac{% 1}{2}\atop a+\frac{1}{2},b+\frac{1}{2},\frac{3}{2}-k-c};1\right)z^{k},}}
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    DLMF:16.12.E3
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