Mathematical Methods in Science and Engineering, 2nd Edition

Chapter 4Gegenbauer and Chebyshev Polynomials

The sine and cosine functions play a central role in the study of oscillations and waves. They come from the solutions of the Helmholtz wave equation in Cartesian coordinates with the appropriate boundary conditions. The sine and cosine functions also form a basis for representing general waves and oscillations of various types, shapes, and sizes. On the other hand, solutions of the angular part of the Helmholtz equation in spherical polar coordinates are the spherical harmonics. Analogous to the oscillations of a piece of string, spherical harmonics correspond to the oscillations of a two-sphere, that is, the surface of a sphere in three dimensions. Spherical harmonics also form a complete set of orthonormal functions; hence they are very important in many theoretical and practical applications. For the oscillations of a three-sphere (hypersphere), along with the spherical harmonics, we also need the Gegenbauer polynomials. Gegenbauer polynomials are very useful in cosmology and quantum field theory in curved backgrounds. Both the spherical harmonics and the Gegenbauer polynomials are combinations of sines and cosines. Chebyshev polynomials form another complete and orthonormal set of functions, which are closely related to the Gegenbauer polynomials.

4.1 Wave Equation on a Hypersphere

Friedmann Robertson Walker models in cosmology, which are also called the standard models, are generally accepted as accurately describing the global ...