On the Integral Representation of Jacobi Polynomials

In this paper, we present a new integral representation for the Jacobi polynomials that follows from Koornwinder’s representation by introducing a suitable new form of Euler’s formula. From this representation, we obtain a fractional integral formula that expresses the Jacobi polynomials in terms of Gegenbauer polynomials, indicating a general procedure to extend Askey’s scheme of classical polynomials by one step. We can also formulate suitably normalized Fourier–Jacobi spectral coefficients of a function in terms of the Fourier cosine coefficients of a proper Abel-type transform involving a fractional integral of the function itself. This new means of representing the spectral coefficients can be beneficial for the numerical analysis of fractional differential and variational problems. Moreover, the symmetry properties made explicit by this representation lead us to identify the classes of Jacobi polynomials that naturally admit the extension of the definition to negative values of the index. Examples of the application of this representation, aiming to prove the properties of the Fourier–Jacobi spectral coefficients, are finally given.