The Addition Formula for Jacobi Polynomials and Spherical Harmonics

Recently Gegenbauer’s addition formula was generalized for Jacobi polynomials Pₙ^( α ,β ) ( x ) by an algebraic approach (cf. the author’s paper in [10]). Here a new algebraic proof using spherical harmonics will be presented. Let q p and let e₁ ,e₂ , ⋯ ,eq + p be an orthonormal basis of Rq + p with unit sphere Ω. The Jacobi polynomials Pₙ( p/2- 1,q/2 - 1) ( x ) can be characterized as spherical harmonics of degree $2n$ on Ω which are invariant under the subgroup SO( q ) × SO( p ) of the rotation group SO( q + p ). Let the rotations A_τ be defined by A_τ eₖ = cos τ eₖ - sin τ eq + k ,k = 1, ⋯ ,q,A_τ eq + k = sin τ eₖ + cos τ eq + k ,k = 1, ⋯ ,q,A_τ eₖ = eₖ ,k = 2q + 1, ⋯ ,q + p. An explicit orthonormal basis will be constructed for the set of those spherical harmonics of degree $2n$ which are invariant under all T ∈ SO( q ) × SO( p ) which commute with the rotations A_τ. Let this basis consist of the functions Sₖ ,k = 1, ⋯ ,N. Then the kernel function Φ (ξ ,η ) = ∑k = 1N Sₖ ( ξ )Sₖ ( η ) ( ξ ,η ∈ Ω ) satisfies Φ ( ξ ,A_τ - 1 e₁ ) = Φ ( A_τ ξ ,e₁ ). The addition formula for Jacobi polynomials Pₙ(p/2 - 1,q/2 - 1) ( x ) is finally obtained by writing this last identity in an explicit way.