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 an approximate complete set of basis, a property implied by their approximate orthogonality.