Gegenbauer, Jacobi, and Orthogonal Polynomials

In earlier chapters we dealt with special sets of orthogonal polynomials, namely, Chebyshev and Hermite polynomials. In Chs. 9 and 10 we will study other orthogonal polynomials, namely, Laguerre and Legendre. All of these polynomial functions share many properties. This indicates that these polynomials are special cases of more general polynomials—Gegenbauer and Jacobi polynomials named after Leopold Gegenbauer (1849–1903) and Carl Gustav Jacob Jacobi (1804–1851). Gegenbauer polynomials are connected with axially symmetric potentials, while Jacobi polynomials are even more general, with Jacobi polynomials containing Gegenbauer polynomials as a special case. Collectively, these polynomials are called classical orthogonal polynomials. In this chapter we look at some of the elementary properties of these polynomials; the reader is referred to other texts for detailed descriptions.