Many functions can be expressed as Meijer-G function with some special values for the parameters: https://en.wikipedia.org/wiki/Meijer_G-function. Gan Heaviside step function and unit step function be expresses as Meijer G function for some parameters?
Motivation: https://papers.nips.cc/paper/9308-demystifying-black-box-models-with-symbolic-metamodels is article about symbolic regression and the innovative use of gradient descent for that. Step functions can be used for the algebraic semantics for logic and hence - if step functions can be expressed via Meijer-G function then this framework can be applied to the finding logical expressions for the use in the symbolic regression.
Btw - encoding of symbolic mathematical expression into the numerical parameters of Meijers-G function is the most innovative and interesting encoding of symbolic expressions I can imagine. Some how tried to use neural networks for that (encoding into the real vectors), but neural networks are not exact encoding. I have never heard about different, alternative encodings of symbolic expressions (apart from 01 encoding used in computer science and encoding using Goedel numbers) but I feel that there should be whole lot theory about that.
I know nothing about the G-functions. But it seems that from Wikipedia we have $$H(|x|-1) = G_{1,1}^{\,0,1} \!\left( \left. \begin{matrix} 1 \\ 0 \end{matrix} \; \right| \, x \right), $$ which is 1 for $|x|>1$ and $0$ for $|x|<1$. If you substitute $x=e^y$ we get 1 for $e^y>1$ i.e. $y>0$ and $0$ for $e^y<1$ i.e. $y<0$. Which is the correct behavior (also at 1), ergo
$$H(x) = G_{1,1}^{\,0,1} \!\left( \left. \begin{matrix} 1 \\ 0 \end{matrix} \; \right| \, e^x \right) $$