  In this paper, a novel class of quantum fractal functions is introduced based
on the Meyer-K\"onig-Zeller operator $M_{q,n}$. These quantum
Meyer-K\"onig-Zeller (MKZ) fractal functions employ $M_{q,n} f$ as the base
function in the iterated function system for $\alpha$-fractal functions. For
$f\in C(I)$, $I$ closed in $\mathbb{R}$, it is shown that there exists a
sequence of quantum MKZ fractal functions
$\{f^{(q_n,\alpha)}_n\}_{n=0}^{\infty}$ which converges uniformly to $f$
without altering the scaling function $\alpha$. The shape of
$f^{(q_n,\alpha)}_n$ depends on $q$ as well as the other IFS parameters. For
$f,g\in C(I)$ with $g > 0$ or $f\geq g$, we show that there exists a sequence
$\{f^{(q_n,\alpha)}_n\}_{n=0}^{\infty}$ with $f^{(q_n,\alpha)}_n \geq g$
converging to $f$. Quantum MKZ fractal versions of some classical M\"untz
theorems are also presented. For $q=1$, the box dimension and some
approximation-theoretic results of MKZ $\alpha$-fractal function are
investigated in $C(I)$. Finally, MKZ $\alpha$-fractal functions are studied in
$L^p$ spaces with ${p \geq 1}$.
