In a previous post I introduced the function $\theta_k(n)$ defined as $$\theta_k(n) = \frac{S(n-k)}{V(n)}$$ where $S(n)$ and $V(n)$ are the surface area and the volume of an $n$-sphere respectively. In the post I presented the identity with the Riemann-zeta function, namely
$$\frac{\zeta (\frac{1}{2}-n)}{\zeta(n+\frac{1}{2})} = -i^{n(n+1)}\frac{\theta_{2n}(2n-1)}{2^n}$$
and conjectured that every $\zeta(2n), n \in \mathbb{N}$ can be written strictly in terms of $\theta_k(n)$ and some power of $\pi$
There are many areas where $\theta_k(n)$ appears, and very often formuls, results, etc can be re-written in terms of $\theta_k(n)$ similar to how $_pF_q(a_1,...,a_p;b_1,...,b_q;z)$ (generalized hypergeometric function) or $\Phi(z, s, a)$ (Lerch transcendant) appear in many results, and are useful in rewriting and simplifying complicated expressions. One example of where a result can be simplified is
$$\sqrt{2\pi}\int_{-\infty}^\infty x^{2n+2}e^{-x^2}\,dx=-\frac{\theta_{2n}(2n+1)}{\left(-\pi\right)^n}$$
One perhaps unsurprising area of mathematics where $\theta_k(n)$ appears is in distributions of vectors on an $n$-sphere. Here are some examples
(Random point uniform on a sphere) The marginal density of randomly uniformly distributed vectors on an $n$-sphere derived from Dirichlet distribution can be written as $$\sim \theta_k(n)\left(1 - \sum_{i=1}^k z_i^2 \right)^{(n-k)/2 -1}$$
Wikipedia link The probability density function, for $y \in [0,1]$ for the square of some first coordinate sampled uniformly at random, can be rewritten as $$\rho(y) = \frac{S(n-1)}{nV(n)}(1-y)^{(n-3)/2}y^{-1/2} = \frac{\theta_1(n)}{n}(1-y)^{(n-3)/2}y^{-1/2}$$
The coefficients in this post can be simplified to
$$\frac{\theta_1(n)}{n}$$
the same appears here
same with (Distribution of correlation of fixed vector on vectors of n-sphere)
Here are some other ones
(calculating a higher order derivative)
(Fourier transform of $1/|x|^{\alpha}$.)
(In how many ways can n couples (husband and wife) be arranged on a bench so no wife would sit next to her husband?) uses an identity of $\theta_{2n}(2n-1)$
Not to mention the multitude of places in which $\theta_{2n}(2n-1)$ appears like in hypergeometric series, Riemann-zeta functions, etc. Is this all just due to the fact that the gamma function is so widely useful? Is there some geometric explanation for its relation to the Riemann-zeta function (for example), or is it all just merely a 'coincidence'? Thanks
