A Generalization of the GGR Conjecture

For each positive integer $n$, function $f$, and point $c$, the GGR Theorem states that $f$ is $n$ times Peano differentiable at $c$ if and only if $f$ is $n-1$ times Peano differentiable at $c$ and the following $n$-th generalized Riemann~derivatives of $f$ at $c$ exist: $$\underset{h\to 0}{\lim }\frac{1}{h^n}\sum _{i=0}^n(-1{)}^i\left(\genfrac{}{}{0px}{}{n}{i}\right)f(c+(n-i-k)h),$$ for $k=0,\dots ,n-1$. The theorem has been recently proved in [AC2] and has been a conjecture by Ghinchev, Guerragio, and Rocca since 1998. We provide a new proof of this theorem, based on a generalization of it that produces numerous new sets of $n$-th Riemann smoothness conditions that can play the role of the above set in the GGR Theorem.