I am new to the Riemann/Siegel Theta function, but it represents many special cases of Inverse Beta Regularized $\text I^{-1}_s(a,b)$. The Riemann theta function can represent any Abelian function, the inverse of any Abelian integral, with the following series expansion. If
$$y=\int_0^x\frac{dt}{\sqrt{\sum\limits_{k=0}^d a_kt^k}}$$ then, its inverse function is a ratio of homogenous polynomials of $$\Theta(\Omega;s)=\Theta\left(\left(\begin{matrix}m_{1,1}&…&m_{1,r}\\\vdots&\ddots&\vdots\\m_{r,1}&…&m_{r,r}\end{matrix}\right);s_1,…,s_r\right)=\sum_{n\in\Bbb Z^r}e^{i\pi(\Omega n^2+2ns)}$$ where $\sum\limits_{n\in\Bbb Z^r}=\sum\limits_{n_1=-\infty}^\infty\cdots\sum\limits_{n_r=-\infty}^\infty$, $a_n$ is a polynomial coefficient, and the dot product is assumed. The inverse beta regularized function is defined by:
$$\text B(a,b)y=\int_0^x t^{a-1} (1-t)^{b-1} dt\implies x=\text I^{-1}_y(a,b);a,b>0,0\le y\le 1$$
Particularly with the Incomplete Beta function $\text B_z(a,b)$:
$$y=\int_0^x\frac1{\sqrt{\left(ax^n+x^m\right)^c}}dx=\frac{\left(-\frac1a\right)^\frac{cn-2}{2(m-n)}}{a^\frac c2(m-n)}\text B_{-\frac{x^{m-n}}a}\left(\frac{cn-2}{2(n-m)},1-\frac c2\right)\implies x=\sqrt[m-n]{-a\operatorname I^{-1}_\frac{a^\frac c2(m-n)y}{\left(-\frac1a\right)^\frac{cn-2}{2(m-n)}\text B\left(\frac{cn-2}{2(n-m)},1-\frac c2\right)} \left(\frac{cn-2}{2(n-m)},1-\frac c2\right)}$$
with $0\le \frac{a^\frac c2(m-n)y}{\left(-\frac1a\right)^\frac{cn-2}{2(m-n)}\text B\left(\frac{cn-2}{2(n-m)},1-\frac c2\right)}\le1,\frac{cn-2}{2(n-m)}>0,c<2$ becoming an Abelian function if $m,n\in\Bbb N,c=0,1,c\to2$
However, there should be more cases where inverse beta regularized is an Abelian function. Note that the Riemann theta function is in Mathematica as SiegelTheta and as RiemannTheta in Maple. Hopefully, we can now extend the $\text I^{-1}_x(a,b)$ function’s domain since $\Theta(\Omega;s)$ can have complex arguments.
What is inverse beta regularized in terms of Riemann theta?
