Gegenbauer Polynomials

Abstract

In the constructive theory of spherical functions the Gegenbauer polynomials play an important role. Apart from constant factors they are certain Jacobi polynomials. For α, β> −1, the indices, the Jacobi polynomials P ^(α,β)_µ of degree μ∈ℕ₀are defined as orthogonal polynomials with respect to the inner product

$$ \left( {f,g} \right): = \int\limits_{ - 1}^1 {f\left( x \right)g\left( x \right){{\left( {1 - x} \right)}^\alpha }{{\left( {1 + x} \right)}^\beta }dx},$$

(2.1)

α, β> -1, which are normalized by the condition

$$ P_\mu ^{\alpha,\beta }\left( 1 \right) = \left( {\begin{array}{*{20}{c}} {\mu + \alpha } \\ \mu \end{array}} \right).$$

(2.2)

.