DLMF:18.11.E3 (Q5637)

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DLMF:18.11.E3
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    Defining formula
    H n ⁡ ( x ) = 2 n ⁢ U ⁡ ( - 1 2 ⁢ n , 1 2 , x 2 ) = 2 n ⁢ x ⁢ U ⁡ ( - 1 2 ⁢ n + 1 2 , 3 2 , x 2 ) = 2 1 2 ⁢ n ⁢ e 1 2 ⁢ x 2 ⁢ U ⁡ ( - n - 1 2 , 2 1 2 ⁢ x ) . Hermite-polynomial-H 𝑛 𝑥 superscript 2 𝑛 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 superscript 𝑥 2 superscript 2 𝑛 𝑥 Kummer-confluent-hypergeometric-U 1 2 𝑛 1 2 3 2 superscript 𝑥 2 superscript 2 1 2 𝑛 superscript 𝑒 1 2 superscript 𝑥 2 parabolic-U 𝑛 1 2 superscript 2 1 2 𝑥 {\displaystyle{\displaystyle H_{n}\left(x\right)=2^{n}U\left(-\tfrac{1}{2}n,% \tfrac{1}{2},x^{2}\right)=2^{n}xU\left(-\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{3}{2% },x^{2}\right)=2^{\frac{1}{2}n}e^{\frac{1}{2}x^{2}}U\left(-n-\tfrac{1}{2},2^{% \frac{1}{2}}x\right).}}
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    DLMF id
    DLMF:18.11.E3
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