$\begin{array}{ll}{\int }_{0}^{\infty }{x}^{2\mu -\nu }{e}^{-\frac{1}{4}{x}^{2}}{M}_{k,\mu }\left(\frac{1}{2}{x}^{2}\right){Y}_{\nu }\left(xy\right)\text{d}x\hfill & ={\pi }^{-1}{2}^{\mu +\beta }{y}^{k-\mu -\frac{3}{2}}{e}^{-\frac{1}{4}{y}^{2}}\mathrm{\Gamma }\left(2\mu +1\right)\hfill \\ ×\mathrm{\Gamma }\left(\frac{1}{2}-k-\mu \right)\left\{\mathrm{cos}\left[\left(\nu -2\mu \right)\pi \right]\frac{\mathrm{\Gamma }\left(2\mu -\nu -1\right)}{\mathrm{\Gamma }\left(2\beta +1\right)}{M}_{\alpha ,\beta }\left(\frac{1}{2}{y}^{2}\right)-\mathrm{sin}\left[\left(\nu +k-\mu \right)\pi \right]{W}_{\alpha ,\beta }\pi \left(\frac{1}{2}{y}^{2}\right)\right\}\hfill \\ \text{?}2\alpha =3\mu -\nu +k+\frac{1}{2},\text{?}\text{?}2\beta =\mu -\nu -k+\frac{1}{2}\hfill \end{array}$ ET II 116(44)

$[y>0,−1<2Reμ−1]$ 14.

$\begin{array}{ll}{\int }_{0}^{\infty }{x}^{2\mu +\nu }{e}^{-\frac{1}{4}{x}^{2}}{M}_{k,\mu }\left(\frac{1}{2}{x}^{2}\right){Y}_{\nu }\left(xy\right)\text{d}x\hfill & ={\pi }^{-1}{2}^{\mu +\beta }{y}^{k-\mu -\frac{3}{2}}\mathrm{\Gamma }\left(2\mu +1\right)\hfill \\ ×\mathrm{\Gamma }\left(\frac{1}{2}-\mu -k\right){e}^{-\frac{1}{4}{y}^{2}}\left\{\mathrm{cos}\left(2\mu \pi \right)\frac{\mathrm{\Gamma }\left(2\mu +\nu +1\right)}{\mathrm{\Gamma }\left(\mu +\nu -k+\frac{3}{2}\right)}{M}_{\alpha ,\beta }\left(\frac{1}{2}{y}^{2}\right)+\mathrm{sin}\left[\left(\mu -k\right)\pi \right]{W}_{\alpha ,\beta }\left(\frac{1}{2}{y}^{2}\right)\right\}\hfill \\ 2\alpha =3\mu +\nu +k+\frac{1}{2},2\beta =\mu +\nu -k+\frac{1}{2}\hfill \end{array}$ ET II 116(43)

$[y>0,−1<2Reμ−1]$

15.

$\begin{array}{l}{\int }_{0}^{\infty }{x}^{2\mu +}\hfill \end{array}$

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