In order to evaluate the convergence, we estimate the decreasing speed of $an$. For representing an odd-order aspherical surface, let $\alpha $ be a noninteger and $\alpha \u22651/2$. The minimum $\alpha =1/2$ corresponds to the first-order surface or the cone. By Euler’s reflection formula $\Gamma (z)\Gamma (1\u2212z)=sin\u2009\pi z\pi $, Display Formula
$\Gamma (\alpha \u2212n+1)=\u2009\u2009sin\u2009\pi (n\u2212\alpha )\pi 1\Gamma (n\u2212\alpha ).$(9)
By the definition of the gamma function of $\Gamma (z+1)=z\Gamma (z)$, Display Formula$\Gamma (\alpha +n+2)=(\alpha +n+1)(\alpha +n)\Gamma (n+\alpha ).$(10)
The denominator of Eq. (6) is Display Formula$\Gamma (\alpha +n+2)\Gamma (\alpha \u2212n+1)=sin\u2009\pi (n\u2212\alpha )\pi (\alpha +n+1)\u2062(\alpha +n)\Gamma (n+\alpha )\Gamma (n\u2212\alpha ).$(11)
Since $\alpha \u22651/2$, the inequality $n+\alpha >n+\alpha \u22121\u2265n\u2212\alpha $ holds. Since $\Gamma (x)>\Gamma (y)$ for positive numbers, $x>y>0$. Then, $\Gamma (n+\alpha )$ is evaluated as Display Formula$\Gamma (n+\alpha )=(n+\alpha \u22121)\Gamma (n+\alpha \u22121)>(n+\alpha \u22121)\Gamma (n\u2212\alpha )where\u2009\u2009n>\alpha .$(12)
Therefore, Display Formula$|\Gamma (\alpha +n+2)\Gamma (\alpha \u2212n+1)|>\u2009\u2009|sin\u2009\pi (n\u2212\alpha )\pi |(\alpha +n+1)(\alpha +n)(n+\alpha \u22121).$(13)
Thus, Display Formula$|an|=|(2n+1)\Gamma (\alpha +1)2\Gamma (\alpha +n+2)\Gamma (\alpha \u2212n+1)|<M|(2n+1)(\alpha +n+1)(\alpha +n)(\alpha +n\u22121)|,$(14)
where $M=\u2009\u2009|sin\u2009\pi (n\u2212\alpha )\pi |$ is a positive number. Hence, $|an|$ decreases faster than or equally to $n\u22122$.