Connection formulas between Fourier–Legendre/Jacobi expansions and central binomial series

In this article, we present several general formulas that establish connections between series involving the Taylor expansion coefficients of a function and series involving the Fourier–Legendre expansion coefficients of the same functions. Consequently, we prove the utility of such formulas for the closed-form evaluation of certain classes of series. Additionally, we illustrate that by leveraging the theory and relationships of Jacobi polynomials, it is possible to derive other compelling identities from the primary connection formulas.