New Proof of the Addition Theorem for Gegenbauer Polynomials

The quantity (λ - z)^ρ is expanded in Jacobi polynomials Pₙ(α ,β ) (z), where α, β, and ρ are unrelated. The known case α = β = - ρ - 1 is then used in a short proof of the addition theorem for Gegenbauer polynomials. The only other ingredients of the proof are well-known generating relations for these polynomials.