Modified Lagrange-Jacobi Functions

This paper presents modified Lagrange-Jacobi functions derived from the sine, exponential, and hyperbolic tangent coordinate transformations. The resulting Lagrange-Jacobi functions and their respective matrix elements for observables can be reduced to their respective Lagrange-Legendre, Lagrange-Chebyshev, and Lagrange-Gegenbauer functions. Furthermore, this paper postulates that the Lagrange-mesh functions form approximate
complete set of basis, a property implied by their approximate orthogonality.