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We prove that a customary Sturm-Liouville form of second-order $q$-differenceequation for the continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ ofRogers 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 thatthe continuous $q$-ultraspherical polynomials $C_n(x;\beta| q)$ are actuallygoverned 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 asquare root of the equation, obtained from its original form.

On factorization of $q$-difference equation for continuous $q$-ultraspherical polynomials