What question did this study set out to answer?

The central aim is to derive modified Lagrange–Jacobi functions and explore their properties and implications.

March 26, 2026Mathematics0 citationsOpen Access

Modified Lagrange–Jacobi Functions

GRG. J. RamphoUniversity of South Africa

Key Points

The central aim is to derive modified Lagrange–Jacobi functions and explore their properties and implications.
Derived functions from sine, exponential, and hyperbolic tangent transformations.
Reduced the functions to Lagrange–Legendre, Lagrange–Chebyshev, and Lagrange–Gegenbauer forms.
Analyzed matrix elements related to observables.
Established modified Lagrange–Jacobi functions as a complete basis set.
Demonstrated approximate orthogonality of the Lagrange-mesh functions.

Abstract

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.

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