Formula:6446

Digital Library of Mathematical Functions ID 18.17.E8

${\displaystyle{\displaystyle\left(H_{n}\left(x\right)\right)^{2}+2^{n}(n!)^{2}e^{x^{2}}\left(V\left(-n-\tfrac{1}{2},2^{\frac{1}{2}}x\right)\right)^{2}=\frac{2^{n+\frac{3}{2}}n!\,e^{x^{2}}}{\pi}\int_{0}^{\infty}\frac{e^{-(2n+1)t+x^{2}\tanh t}}{(\sinh 2t)^{\frac{1}{2}}}\mathrm{d}t.}}$

Symbols List

• ${\displaystyle{\displaystyle H_{\NVar{n}}\left(\NVar{x}\right)}}$: Hermite polynomial
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\tanh\NVar{z}}}$: hyperbolic tangent function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle V\left(\NVar{a},\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle n}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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