Integral Representation Technique for Expansion of Arbitrary Analytic Functions of the Distance between Two Points

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Abstract
The coefficients of expansion of an arbitrary analytic function of the distance r between two points (r1, θ1, φ1) and (r2, θ2, φ2) in terms of the Legendre polynomials Pl(cos θ12) are double Bessel transformed. Assuming that the transformed coefficients are diagonal, consistent with the differential equations satisfied by the original coefficients, we derive the explicit expressions for the latter coefficients. These formulas are identical to those derived by Sack from the solutions of the differential equations in terms of the hypergeometric functions.

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