DLMF:12.13.E6 (Q4235)

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DLMF:12.13.E6
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    n ! ⁢ U ⁡ ( n + 1 2 , z ) = i n ⁢ e - 1 2 ⁢ z 2 ⁢ erfc ⁡ ( z / 2 ) ⁢ U ⁡ ( - n - 1 2 , i ⁢ z ) + ∑ m = 1 ⌊ 1 2 ⁢ n + 1 2 ⌋ U ⁡ ( 2 ⁢ m - n - 1 2 , z ) , 𝑛 parabolic-U 𝑛 1 2 𝑧 superscript 𝑖 𝑛 superscript 𝑒 1 2 superscript 𝑧 2 complementary-error-function 𝑧 2 parabolic-U 𝑛 1 2 𝑖 𝑧 superscript subscript 𝑚 1 1 2 𝑛 1 2 parabolic-U 2 𝑚 𝑛 1 2 𝑧 {\displaystyle{\displaystyle n!U\left(n+\tfrac{1}{2},z\right)=i^{n}e^{-\frac{1% }{2}z^{2}}\operatorname{erfc}(z/\sqrt{2})U\left(-n-\tfrac{1}{2},iz\right)+\sum% _{m=1}^{\left\lfloor\frac{1}{2}n+\frac{1}{2}\right\rfloor}U\left(2m-n-\tfrac{1% }{2},z\right),}}
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    DLMF:12.13.E6
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