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## 7.54 Combinations of hypergeometric and Bessel functions

7.541

${\int }_{0}^{\infty }{x}^{\alpha +\beta -2\nu -1}{\left(x+1\right)}^{-\nu }{e}^{xz}{K}_{\nu }\left[\left(x+1\right)z\right]F\left(\alpha ,\beta ;\alpha +\beta -2\nu ;-x\right)\text{d}x={\pi }^{-\frac{1}{2}}\mathrm{cos}\left(\nu \pi \right)\mathrm{\Gamma }\left(\frac{1}{2}-\alpha +\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-\beta +\nu \right)\mathrm{\Gamma }\left(\gamma \right){\left(2z\right)}^{-\frac{1}{2}-\frac{1}{2}\gamma }{W}_{\frac{1}{2}\lambda ,\frac{1}{2}\left(\beta -\alpha \right)\left(2z\right)}$

ET II 401(16)

$γ=α+β−2ν[Re(α+β−2ν)>0,Re(12−α+υ)>0,Re(12−β+υ)>0,|argz|<32π]$

7.542

1.12

${\int }_{0}^{\infty }{x}^{\sigma -1}F_{p}{p-1}_{}\left({a}_{1},\dots ,{a}_{p};{b}_{1},\dots ,{b}_{p-1};-\lambda {x}^{2}\right){Y}_{\nu }\left(xy\right)\text{d}x=\frac{\mathrm{\Gamma }\left({b}_{1}\right)\dots \mathrm{\Gamma }\left({b}_{p-1}\right)}{2{\lambda }^{\frac{1}{2}\sigma }\mathrm{\Gamma }\left({a}_{1}\right)\dots \mathrm{\Gamma }\left({a}_{p}\right)}{G}_{p+2,p+3}^{p+2,1}\left(\frac{{y}^{2}}{4\lambda }|\begin{array}{c}{b}_{0}^{*},\dots ,{b}_{p-1}^{*},l\\ h,k,{a}_{1}^{*},\dots ,{a}_{p}^{*},l\end{array}\right)$

ET II 118(53)

$aj*=aj−σ2,j=1,…,p;b0*=1−σ2;bj*=bj−σ2,j=1,…,p$

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