  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.
