Formula:KLS:09.08:42

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2 Γ ⁡ ( λ + 1 2 ) ∫ 0 1 C n λ ⁡ ( t ) C n λ ⁡ ( 1 ) ( 1 - t 2 ) λ - 1 2 t - 1 ( x / t ) n + 2 ⁢ λ + 1 e - x 2 / t 2 d t = 2 - n H n ⁡ ( x ) e - x 2 ( λ > - 1 2 ) fragments 2 Euler-Gamma 𝜆 1 2 superscript subscript 0 1 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 𝑡 ultraspherical-Gegenbauer-polynomial 𝜆 𝑛 1 superscript fragments ( 1 superscript 𝑡 2 ) 𝜆 1 2 superscript 𝑡 1 superscript fragments ( x t ) 𝑛 2 𝜆 1 superscript 𝑥 2 superscript 𝑡 2 d t superscript 2 𝑛 Hermite-polynomial-H 𝑛 𝑥 superscript 𝑥 2 fragments ( λ 1 2 ) {\displaystyle{\displaystyle{\displaystyle\frac{2}{\Gamma\left(\lambda+\frac{1% }{2}\right)}\int_{0}^{1}\frac{C^{\lambda}_{n}\left(t\right)}{C^{\lambda}_{n}% \left(1\right)}(1-t^{2})^{\lambda-\frac{1}{2}}t^{-1}(x/t)^{n+2\lambda+1}{% \mathrm{e}^{-x^{2}/t^{2}}}dt=2^{-n}H_{n}\left(x\right){\mathrm{e}^{-x^{2}}}(% \lambda>-\frac{1}{2})}}}

Proof

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Symbols List

Γ Γ {\displaystyle{\displaystyle{\displaystyle\Gamma}}}  : Euler's gamma function : http://dlmf.nist.gov/5.2#E1
∫ {\displaystyle{\displaystyle{\displaystyle\int}}}  : integral : http://dlmf.nist.gov/1.4#iv
C n μ subscript superscript 𝐶 𝜇 𝑛 {\displaystyle{\displaystyle{\displaystyle C^{\mu}_{n}}}}  : ultraspherical/Gegenbauer polynomial : http://dlmf.nist.gov/18.3#T1.t1.r5
e e {\displaystyle{\displaystyle{\displaystyle\mathrm{e}}}}  : the base of the natural logarithm : http://dlmf.nist.gov/4.2.E11
H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : Hermite polynomial H n subscript 𝐻 𝑛 {\displaystyle{\displaystyle{\displaystyle H_{n}}}}  : http://dlmf.nist.gov/18.3#T1.t1.r28

Bibliography

Equation in Section 9.8 of KLS.

URL links

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