New linearization formula for $q$-ultraspherical polynomials

For the product $C_m(x;\beta| q)C_n(x;\beta'| q)$ of two continuous $q$-ultraspherical polynomials with different parameters $\beta$ and $\beta'$, explicit formulae are presented for the coefficients of its expansion in the basis $C_k(x;\beta| q)$ in the case $\beta'=q\beta-1$. We also discuss a generalization to multivariate Jack and Macdonald polynomials and its relation to Pieri formulas and $Q$-operators.