We provide the first two examples of sets of generalized Riemann derivatives of orders up to n, \(n\ge 2\) , whose simultaneous existence for all functions f at x is equivalent to the existence of the n-th Peano derivative \(f_{(n)}(x)\) . In this way, we begin to understand how the theory of Peano derivatives can be explained exclusively in terms of generalized Riemann derivatives, a bold new principle in generalized differentiation. In 1936, J. Marcinkiewicz and A. Zygmund showed that the existence of \(f_{(n)}(x)\) is equivalent to the existence of both \(f_{(n-1)}(x)\) and the nth generalized Riemann derivative \(\widetilde{D}_nf(x)\) , based at \(x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h\) . Our first characterization of \(f_{(n)}(x)\) is that its existence is equivalent to the simultaneous existence of \(\widetilde{D}_1f(x),\ldots ,\widetilde{D}_nf(x)\) . Our second characterization is that the existence of \(f_{(n)}(x)\) is equivalent to the existence of \(\widetilde{D}_1f(x)\) and of all \(n(n-1)/2\) forward shifts, \(\begin{aligned} D_{k,j}f(x)=\lim _{h\rightarrow 0} h^{-k}\sum _{i=0}^k(-1)^i\left( {\begin{array}{c}k\\ i\end{array}}\right) f(x+(k+j-i)h), \end{aligned}\) for \(j=0,1,\ldots ,k-2\) , of the k-th Riemann derivatives \(D_{k,0}f(x)\) , for \(k=2,\ldots ,n\) . The proof of the second result involves an interesting combinatorial algorithm that starts with consecutive forward shifts of an arithmetic progression and yields a geometric progression, using two set-operations: dilation and combinatorial Gaussian elimination. This result proves a variant of a 1998 conjecture by Ginchev, Guerragio and Rocca, predicting the same outcome for backward shifts instead of forward shifts. The conjecture has been recently settled in Ash and Catoiu (Int Math Res Not IMRN 10:7893–7921, 2022), with a proof that has this variant’s proof as a prerequisite.
