Gaussian Riemann derivatives

J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions f points x, the existence of nth Peano derivative f(n)(x) is equivalent to both f(n−1)(x) generalized Riemann $${{\tilde D}_n}f\left(x \right)$$ , based at x,x + h,x 2h,x 22{h,…,x} 2n−1h. For q ≠ 0, ±1, we introduce: two q-analogues n-th Dnf(x) Gaussian derivatives qDnf(x) $$_q{{\bar are h, x+qh, x+q2h,…, x qn−1h x+h,x qh, q2h,…,x+qnh; one analogue symmetric $$D_n^sf\left(x $$_qD_n^sf\left(x (x), x±h, x±qh, x±q2h, …, x±qm−1h, where m = ⌊(n+1)/2⌋ (x) means that taken only n even. We provide exact expressions their associated differences terms binomial coefficients; show satisfy above classical theorem, satisfies a version theorem; conjecture these results false every larger classes derivatives, thereby extending recent conjectures by Ash Catoiu, which update answering them few cases.