Mixed series in ultraspherical polynomials and their approximation properties

New (mixed) series in ultraspherical polynomials $P_n^{\alpha,\alpha}(x)$ are introduced. The basic difference between a mixed series in the polynomials $P_n^{\alpha,\alpha}(x)$ and a Fourier series in the same polynomials is as follows: a mixed series contains terms of the form $\dfrac{2^rf_{r,k}^\alpha}{(k+2\alpha)^{[r]}}P_{k+r}^{\alpha-r,\alpha-r}(x)$, where $1\leqslant r$ is an integer and $f_{r,k}^\alpha$ is the $k$ th Fourier coefficient of the derivative $f^{(r)}(x)$ with respect to the ultraspherical polynomials $P_k^{\alpha,\alpha}(x)$. It is shown that the partial sums ${\mathscr Y}_{n+2r}^\alpha(f,x)$ of a mixed series in the polynomial $P_k^{\alpha,\alpha}(x)$ contrast favourably with Fourier sums $S_n^\alpha(f,x)$ in the same polynomials as regards their approximation properties in classes of differentiable and analytic functions, and also in classes of functions of variable smoothness. In particular, the ${\mathscr Y}_{n+2r}^\alpha(f,x)$ can be used for the simultaneous approximation of a function $f(x)$ and its derivatives of orders up to $(r- 1)$, whereas the $S_n^\alpha(f,x)$ are not suitable for this purpose.