Criterion of square summability with geometric weight for Jacobi expansions

In this paper we prove that $\sum_{k=0}^{\infty}|f_k|^2\theta^k<\infty$, where $\theta>1$ and $f_k$ is the k-th Fourier coefficient of a function $f\in{L_1(-1,1;(1-x)^{\lambda}(1+x)^{\mu})}$ in orthonormal Jacobi polynomials, iff $f$ can be analytically continued to the ellipse $E_{\theta}=\{z:~|z-1|+|z+1|<\theta^{\frac{1}{2}}+ \theta^{-\frac{1}{2}}\}$ and its analytic continuation belongs to the Szegö space $AL_2(\partial{E_{\theta}})$.