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