The Paneitz operator in even dimensional sphere is defined by $$ P_n=\prod_{k=0}^{\frac{n-2}{2}}(-\Delta+k(n-k-1)), $$ where $\Delta$ is the laplacian on sphere with usual metric $g_{S^n}=(1-x^2)^{-1}+(1-x^2)g_{S^{n-1}}$.
Consider a axially symmetric function on the sphere, say $u=u(x)$. Clearly, $$ P_2(u)=-\Delta u=|g|^{-1/2}\frac{\partial}{\partial x}(|g|^{1/2}g^{11}\frac{\partial u}{\partial x})=(1-x^2)^{-n/2+1}\frac{\partial}{\partial x}((1-x^2)^{n/2-1}(1-x^2)\frac{\partial u}{\partial x})\\ =(1-x^2)u''-nxu'=-[(1-x^2)u']'. $$
Direct computation then shows $$ P_4(u)=[(1-x^2)u']^{(3)}. $$
In this paper Improved Beckner's inequality for axially symmetric functions on $S^n$, the author claims when n is even, $$ P_n(u)=(-1)^{n/2}[(1-x^2)u']^{(n-1)}, $$ but I have no idea how to prove this. I have tried to use induction and write $$ P_{n+2}=\prod_{k=0}^{\frac{n}{2}}(-\Delta+k(n-k+1))=\prod_{k=1}^{\frac{n}{2}}(-\Delta+k(n-k+1))(-\Delta)=\prod_{k=0}^{\frac{n-2}{2}}(-\Delta+(k+1)(n-k))(-\Delta)\\=\prod_{k=0}^{\frac{n-2}{2}}(-\Delta+k(n-k-1)+n)(-\Delta), $$ but I do not know how to continue.
This question should be simple and dose not rely on additional advance knowledge. Appreciate any help!