DLMF:12.7.E3 (Q4126)

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DLMF:12.7.E3
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    Defining formula
    V ⁡ ( n + 1 2 , z ) = 2 / π ⁢ e 1 4 ⁢ z 2 ⁢ ( - i ) n ⁢ 𝐻𝑒 n ⁡ ( i ⁢ z ) = 2 / π ⁢ e 1 4 ⁢ z 2 ⁢ ( - i ) n ⁢ 2 - 1 2 ⁢ n ⁢ H n ⁡ ( i ⁢ z / 2 ) , parabolic-V 𝑛 1 2 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑖 𝑛 Hermite-polynomial-He 𝑛 𝑖 𝑧 2 𝜋 superscript 𝑒 1 4 superscript 𝑧 2 superscript 𝑖 𝑛 superscript 2 1 2 𝑛 Hermite-polynomial-H 𝑛 𝑖 𝑧 2 {\displaystyle{\displaystyle V\left(n+\tfrac{1}{2},z\right)=\sqrt{2/\pi}e^{% \frac{1}{4}z^{2}}(-i)^{n}\mathit{He}_{n}\left(iz\right)=\sqrt{2/\pi}e^{\frac{1% }{4}z^{2}}(-i)^{n}2^{-\frac{1}{2}n}H_{n}\left(iz/\sqrt{2}\right),}}
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    DLMF id
    DLMF:12.7.E3
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