A novel method which has been developed in Demiralp's group and has been named "Separate Node Ascending Derivatives Expansion (SNADE) for Univariate Functions" is the content of the first one of four companion papers to appear in this proceedings. This work focuses on the construction of basic recursions amongst the SNADE polynomials and also construction of a bound to SNADE remainder term. These goals are achieved. The convergence of SNADE could have also been proven by using the derived remainder bound. The recursions are constructed in both implicit and explicit form.