Using quantum field theory arguments, in https://arxiv.org/pdf/2401.05733.pdf, we found that there is a parametric representation of the (slightly generalised) Euler-Beta function:
\begin{equation}\label{gen-beta} \frac{\Gamma\left(\alpha-s_{1}\right)\Gamma\left(\alpha-s_{2}\right)}{\Gamma\left(\beta-s_{1}-s_{2}\right)} =\sum_{n=0}^{\infty}\frac{(-1)^{p+1}}{n!}\left(\frac{1}{s_{1}-\alpha-n}+\frac{1}{s_{2}-\alpha-n}+\frac{1}{\alpha+\lambda+n}\right) %\frac{\sin\left[\pi\left(\alpha-\beta-\lambda+\frac{\left(s_{1}+\lambda\right)\left(s_{2}+\lambda\right)}{\alpha+\lambda+n}\right)\right]}{\sin\left[\pi\left(\alpha+\lambda-\frac{\left(s_{1}+\lambda\right)\left(s_{2}+\lambda\right)}{\alpha+\lambda+n}\right)\right]}\nonumber\\ \left(1-\alpha-\lambda+\frac{\left(s_{1}+\lambda\right)\left(s_{2}+\lambda\right)}{\alpha+\lambda+n}\right)_{n+p} \end{equation} Here $2\alpha-\beta=p$. $(a)_b=\Gamma(a+b)/\Gamma(a)$ is the Pochhammer symbol or rising factorial. The standard Euler-Beta can be reached by setting $\alpha=\beta=0$. Here $\lambda \in \mathbb{C}$ is a parameter such that $\Re(\lambda) > 0$.
We are unaware of similar representations in the maths literature. As corollaries to the above result, one can easily find parametric representations of the Zeta function as well as $\pi$. For instance we have:
\begin{eqnarray} \pi=4+\sum_{n=1}^\infty \frac{1}{n!}\left(\frac{1}{n+\lambda}-\frac{4}{2n+1}\right)\left(\frac{(2n+1)^2}{4(n+\lambda)}-n\right)_{n-1}.\nonumber \end{eqnarray}
Curiously, the summand for large $\lambda$, fixed $n$ goes over to the Madhava-Leibniz summand $(-1)^n 4/(2n+1)$ and by adjusting $\lambda$ one can get faster convergence. The goal of all this was not to come up with fast converging answers, of course!
There are parametric representations of Zeta and $1/pi$ but none of them appear to be systematic. We could not find anything for the Euler-Beta. Are any of these formulas displayed above known in the mathematics literature?