ScalingStacks

Proposition 5.3 . [024F]

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Proposition 5.3.

Let n≥3n\geq 3. Suppose gg satisfies the ODE

dd​y​(gy)=1y2​(1−yn)12,0<y<1,\frac{d}{dy}\left(\frac{g}{y}\right)=\frac{1}{y^{2}(1-y^{n})^{\frac{1}{2}}},\qquad 0<y<1,

together with the initial conditions g⁡(0)=−1,g′​(0)=0g(0)=-1,g^{\prime}(0)=0. Then we have

g(y)=−2F1[12,−1n;n−1n;yn].g(y)=-\,_{2}F_{1}[\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};y^{n}].

In particular

g(1)=limy→1g(y)=−2F1[12,−1n;n−1n;1]=−Γ⁡(n−1n)​πΓ⁡(n−22​n).g(1)=\lim_{y\to 1}g(y)=-\,_{2}F_{1}[\frac{1}{2},-\frac{1}{n};\frac{n-1}{n};1]=-\frac{\Gamma(\frac{n-1}{n})\sqrt{\pi}}{\Gamma(\frac{n-2}{2n})}.

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