Gaussian Riemann Derivatives

Abstract

J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions $f$ and points $x$, the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann derivative $\widetilde{D}_nf(x)$, based at $x,x+h,x+2h,x+2^2h,\ldots ,x+2^{n-1}h$. For $q\neq 0,\pm 1$, we introduce: two $q$-analogues of the $n$-th Riemann derivative ${D}_nf(x)$ of~$f$ at~$x$, the $n$-th Gaussian Riemann derivatives ${_q}{D}_nf(x)$ and ${_q}{\bar D}_nf(x)$ are the $n$-th generalized Riemann derivatives based at $x,x+h,x+qh,x+q^2h,\ldots ,x+q^{n-1}h$ and $x+h,x+qh,x+q^2h,\ldots ,x+q^{n}h$; and one analog of the $n$-th symmetric Riemann derivative ${D}_n^sf(x)$, the $n$-th symmetric Gaussian Riemann derivative ${_q}{D}_n^sf(x)$ is the $n$-th generalized Riemann derivative based at $(x),x\pm h,x\pm qh,x\pm q^2h,\ldots ,x\pm q^{m-1}h$, where $m=\lfloor (n+1)/2\rfloor $ and~$(x)$ means that $x$ is taken only for $n$ even. We provide the exact expressions for their associated differences in terms of Gaussian binomial coefficients; we show that the two $n$th Gaussian derivatives satisfy the above classical theorem, and that the $n$th symmetric Gaussian derivative satisfies a symmetric version of the theorem; and we conjecture that these two results are false for every larger classes of generalized Riemann derivatives, thereby extending two recent conjectures by Ash and Catoiu, both of which we update by answering them in a few cases.