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

We prove that a customary Sturm-Liouville form of second-order $q$-difference equation for the continuous $q$-ultraspherical polynomials $C_n(x;\beta \mathrm{\mid }q)$ 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 $C_n(x;\beta \mathrm{\mid }q)$ 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.