The hyperserial field of surreal numbers

For any ordinal $\alpha >0$, we show how to define a hyperexponential $E_{{\omega }^{\alpha }}$ and a hyperlogarithm $L_{{\omega }^{\alpha }}$ on the class ${\mathbf{N}\mathbf{o}}^{>,\succ }$ of positive infinitely large surreal numbers. Such functions are archetypes of extremely fast and slowly growing functions at infinity. We also show that the surreal numbers form a so-called hyperserial field for our definition.