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For a one-parameter family of lower triangular matrices with entriesinvolving continuous $q$-ultraspherical polynomials we give an explicit lowertriangular inverse matrix, with entries involving again continuous$q$-ultraspherical functions. The matrices are $q$-analogues of results givenby Cagliero and Koornwinder recently. The proofs are not $q$-analogues of theCagliero-Koornwinder case, but are of a different nature involving $q$-Racahpolynomials. Some applications of these new formulas are given. Also the limit$\beta \to 0$ is studied and gives rise to continuous $q$-Hermite polynomialsfor $0 < q < 1$ and $q > 1$.

Explicit matrix inverses for lower triangular matrices with entries involving continuous q-ultraspherical polynomials