On a transcendental equation involving quotients of Gamma functions

Luo, Senping; Wei, Juncheng; Zou, Wenming

doi:10.1090/proc/13408

On a transcendental equation involving quotients of Gamma functions
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by Senping Luo, Juncheng Wei and Wenming Zou PDF

Proc. Amer. Math. Soc. 145 (2017), 2623-2637 Request permission

Abstract:

This note is aimed at giving a complete characterization of the following equation in $p$: \[ \displaystyle p\frac {\Gamma (\frac {n}{2}-\frac {s}{p-1})\Gamma (s+\frac {s}{p-1})}{\Gamma (\frac {s}{p-1})\Gamma (\frac {n-2s}{2}-\frac {s}{p-1})} =\Big (\frac {\Gamma (\frac {n+2s}{4})}{\Gamma (\frac {n-2s}{4})}\Big )^2.\]

The method is based on some key transformations and the properties of the Gamma function. Applications to fractional nonlinear Lane-Emden equations will be given.

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