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Lemma A.3 .
The following asymptotics hold:
(1)
Let α ∈ ℝ ∖ { 0 , − 1 , − 2 , − 3 , … } \fa\in\mathbb{R}\setminus\{0,-1,-2,-3,\ldots\} and β > 0 \fb>0 satisfy α > β + 1 \fa>\fb+1 , then
(A.25) Φ ♯ ( β , α , y ) ∼ { Γ ( α ) Γ ( α − β ) ⋅ ( − y ) − β , y → − ∞ , Γ ( α ) Γ ( β ) ⋅ e y ⋅ y β − α , y → + ∞ . \displaystyle\Ku(\fb,\fa,y)\sim\begin{cases}\frac{\Gamma(\alpha)}{\Gamma(\alpha-\beta)}\cdot(-y)^{-\beta},&y\to-\infty,\\
\frac{\Gamma(\fa)}{\Gamma(\fb)}\cdot e^{y}\cdot y^{\fb-\fa},&y\to+\infty.\end{cases}
(2)
Let β > 0 \beta>0 , then
(A.26) 𝒰 ( β , α , y ) ∼ y − β , y → + ∞ . \mathcal{U}(\fb,\fa,y)\sim y^{-\fb},\ y\to+\infty.
(3)
Let α > β \alpha>\beta , then
(A.27) Ψ ♭ ( β , α , y ) ∼ e y ⋅ ( − y ) β − α , y → − ∞ . \Tri(\fb,\fa,y)\sim e^{y}\cdot(-y)^{\fb-\fa},\ y\to-\infty.