  We prove that a customary Sturm-Liouville form of second-order $q$-difference
equation for the continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ of
Rogers can be written in a factorized form in terms of some explicitly defined
$q$-difference operator ${\mathcal D}_x^{\beta, q}$. This reveals the fact that
the continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ are actually
governed by the $q$-difference equation ${\mathcal D}_x^{\beta, q} C_n(x;\beta|
q)= (q^{-n/2}+\beta q^{n/2}) C_n(x;\beta| q)$, which can be regarded as a
square root of the equation, obtained from its original form.
