$\underset{0}{\overset{\infty }{\int }}{x}^{2\lambda -1}{\left(a+x\right)}^{-\mu -\frac{1}{2}}{e}^{-\frac{1}{2}x}{M}_{{k,\mu }_{}}^{{-\frac{1}{2}x}^{}}\left(a+x\right)\text{d}x=\frac{\mathrm{\Gamma }\left(2\lambda \right)\mathrm{\Gamma }\left(2\mu +1\right)\mathrm{\Gamma }\left(k+\mu -2\lambda +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(k+\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(1-2\lambda +2\mu \right)}{a}^{\lambda -\mu -\frac{1}{2}}{M}_{k-\lambda ,\mu -\lambda }\left(a\right)$

ET II 405(20)

3.

$\underset{0}{\overset{\infty }{\int }}{x}^{2\lambda -1}{\left(a+x\right)}^{-\mu -\frac{1}{2}}{e}^{-\frac{1}{2}x}{W}_{k,\mu }\left(a+x\right)\text{d}x=\mathrm{\Gamma }\left(2\lambda \right){a}^{\lambda -\mu -\frac{1}{2}}{W}_{k-\lambda ,\mu -\lambda }\left(a\right)$

ET II 411(47)

4.

$\underset{0}{\overset{\infty }{\int }}{x}^{\lambda -1}{\left(a+x\right)}^{k-\lambda -1}{e}^{-\frac{1}{2}x}{W}_{k,\mu }\left(a+x\right)\text{d}x=\mathrm{\Gamma }\left(\lambda \right){a}^{k-1}{W}_{k-\lambda ,\mu }\left(a\right)$

ET II 411(48)

5.

$\underset{0}{\overset{\infty }{\int }}{x}^{}$

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