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Lemma A.7 .
Assume that α > β \fa>\fb y ≤ 0 y\leq 0
(A.37) Φ ♯ ( β , α , y ) = Γ ( α ) Γ ( α − β ) ⋅ e y ( − y ) 1 − α 2 ⋅ ∫ 0 ∞ e − t ⋅ t α − 1 2 − β ⋅ I α − 1 ( 2 − yt ) dt . \Ku(\fb,\fa,y)=\frac{\Gamma(\fa)}{\Gamma(\fa-\fb)}\cdot e^{y}(-y)^{\frac{1-\fa}{2}}\cdot\int_{0}^{\infty}e^{-t}\cdot t^{\frac{\fa-1}{2}-\fb}\cdot I_{\fa-1}(2\sqrt{-yt})dt.