Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Lemma A.5 .
For any α ∈ ℝ ∖ { 0 , − 1 , − 2 , − 3 , … } \fa\in\mathbb{R}\setminus\{0,-1,-2,-3,\ldots\} β ∈ ℝ \beta\in\mathbb{R} α > β \fa>\fb
(A.34) 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}