Is Chebyshev series with analytic coefficients analytic?

Question

I have a Chebyshev series $\sum_{k_2 = 0}^\infty c_{k_2}(x_1) \, T_{k_2} (x_2)$ (with $T_k(x)$ the Chebyshev polynomials) that converges to an analytic function $f(x_1,x_2)$ on $[-1,1]^2$. If I know that each $c_{k_2}(x_1)$ is analytic on some domain $\Omega \supset [-1,1]$, can I conclude that $f(x_1,x_2)$ is analytic on $\Omega \times [-1,1]$?