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Abstract. We prove that a customary Sturm-Liouville form of second-order q-difference equation for the continuous q-ultraspherical polynomials Cn(x; β | q) of Rogers can be written in a factorized form in terms of some explicitly defined q-difference β, q operator Dx. This reveals the fact that the continuous q-ultraspherical polynomi-β, q als Cn(x; β | q) are actually governed by the q-difference equation Dx Cn(x; β | q) = −n/2 n/2 q + β q) Cn(x; β | q), which can be regarded as a square root of the equation, obtained from its original form. 1

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