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Lemma A.8 .
Let ν > 0 \nu>0 y > 0 y>0
(A.42) I ν ( y ) \displaystyle I_{\nu}(y) = ( y 2 ) ν e − y Γ ( ν + 1 ) Φ ♯ ( ν + 1 2 , 2 ν + 1 , 2 y ) , \displaystyle=\frac{(\frac{y}{2})^{\nu}e^{-y}}{\Gamma(\nu+1)}\Ku(\nu+\frac{1}{2},2\nu+1,2y),
(A.43) K ν ( y ) \displaystyle K_{\nu}(y) = π ( 2 y ) ν e − y 𝒰 ( ν + 1 2 , 2 ν + 1 , 2 y ) . \displaystyle=\sqrt{\pi}(2y)^{\nu}e^{-y}\mathcal{U}(\nu+\frac{1}{2},2\nu+1,2y).