  This paper provides a rigorous and delicate analysis for exponential decay of
Jacobi polynomial expansions of analytic functions associated with the
Bernstein ellipse. Using an argument that can recover the best estimate for the
Chebyshev expansion, we derive various new and sharp bounds of the expansion
coefficients, which are featured with explicit dependence of all related
parameters and valid for degree $n\ge 1$. We demonstrate the sharpness of the
estimates by comparing with existing ones, in particular, the very recent
results in [38, SIAM J. Numer. Anal., 2012]. We also extend this argument to
estimate the Gegenbauer-Gauss quadrature remainder of analytic functions, which
leads to some new tight bounds for quadrature errors.
