On Jacobi Spectral Polynomials and Evaluation of Definite Integrals

In this paper, we construct a new class of spectral polynomials associated with the product P(α,β;γ,δ) k,m := P(α,β) k × P(γ,δ) m  (k, m ≥ 0; α,β,γ,δ > −1), where P(ν,µ) j are the normalised Jacobi polynomials with j ≥ 0;ν,µ > −1. These spectral polynomials are then used to evaluate certain definite integrals involving the combination of logarithmic, exponential and trigonometric functions. These integrals, which are interesting in their own right, are expressed explicitly as polynomials in the eigenvalues λα,β k and λγ,δ m , where λν,µ j := j(j +ν +µ+1),j ≥ 0.