Some Formulas for Legendre Functions Induced by the Poisson Transform

Using the Poisson transform, which maps any homogeneous and infinitely differentiable function on a cone into a corresponding function on a hyperboloid, we derive some integral representations of the Legendre functions.


Introduction
Let us assume that the linear space R n+1 is endowed with the quadratic form q(x) := x 2 0 − x 2 1 − . . .− x 2 n .We denote the polar bilinear form for q by q.The Lorentz group SO(n, 1) preserves this form and divides R n+1 into orbits.We will deal with two kinds of these orbits.One of them is it is a cone.The second kind of orbits consist of two-sheet hyperboloids H(r) := {x | q(x) = r 2 } for any r > 0.
The group SO(n, 1) has 2 connected components.One of them contains the identity and will be under our consideration further.We denote this subgroup by symbol G.The action x −→ g −1 x of the group G is transitive on C. Let σ ∈ C and D σ be a linear subspace in C ∞ (C) consisting of σ-homogeneous functions.It is useful to suppose throughout this paper that −n + 1 < re σ < 0. We define the representation T σ in D σ by left shifts: Suppose that γ is a contour on C intersecting all generatrices (i.e.all lines containing the origin).Every point x ∈ γ depends on n−1 parameters, so every point x ∈ C can be represented as Denoting by G the subgroup of G which acts transitively on γ, we have where dγ is the G-invariant measure on γ.
For any pair (D σ , D σ), we define the bilinear functionals as the Poisson transform [1].
2 Formulas related to sphere and paraboloid Let γ 1 be the intersection of the cone C and the plane x 0 = 1.Each point x ∈ γ 1 depends on spherical parameters φ 1 , . . ., φ n−1 by the formula The research presented in this paper was supported by grant NK 586P-30 from the Ministry of Education and Science of the Russian Federation.
if angle φ n−s+1 exists.Here The subgroup H 1 SO(n) acts transitively on γ 1 , and any permutate ζ ∈ S n+1 defines the H 1invariant measure The invariant measure in spherical coordinates is given by 9.1.1.(9)[2] Let γ 2 be the intersection of cone C and the hyperplane x 0 +x n = 1.We describe every point x ∈ γ 2 by the coordinates r, φ 1 , . . ., φ n−2 according to the formulas We denote as H 2 the subgroup of G acting transitively on γ 2 .H 2 consists of the matrices , We will now deal with two bases in D σ .One of them consists of the functions The second basis consists of the functions where r 2 j = x 2 1 + . . .+ x 2 j , L = (l 1 , . . ., l n−3 , ±l n−2 ) ∈ Z n−2 , λ ≥ 0 and l i ≥ l i+1 ≥ 0. Suppose, in addition, that the functions of the above bases are equipped with the normalizing factors defined by formulas [2, 9.4.1.7,10.3.4.9].
Let us consider the distribution From the orthogonality of the functions Ξ n T , we obtain the property From this property, it immediately follows that ).
Let γ = γ 1 .Then from the formula Let us assume another situation.
Proof.Suppose γ = γ 2 .Then we obtain the integral which can be solved explicitly after replacing Theorem 1.