Alternative Jacobi Polynomials and Orthogonal Exponentials

Sequences of orthogonal polynomials that are alternative to the Jacobi polynomials on the interval $[0,1]$ are defined and their properties are established. An $(\alpha ,\beta )$-parameterized system of orthogonal polynomials of the exponential function on the semi-axis $[0,\mathrm{\infty })$ is presented. Two subsystems of the alternative Jacobi polynomials, as well as orthogonal exponential polynomials are described. Two parameterized systems of discretely almost orthogonal functions on the interval $[0,1]$ are introduced.