Recursive Method for the Computation of the SOn, SO n,1, and ISOn, Representation Matrix Elements

We find a procedure whereby the matrix elements of the finite SOn,1 transformations (principal series) can be expressed as a single integral, over a compact domain, of two matrix elements of the SOn subgroup and a multiplier. In this way we automatically obtain their classification by the canonical chain SOn, 1⊃SOn⊃⋯⊃SO2. Analytic continuation yields the SOn+1 matrix elements in a recursive form. We obtain the asymptotic behavior of the boost matrix elements. The Inönü-Wigner contraction yields the ISOn representation matrix elements classified by the chain ISOn⊃SOn⊃⋯⊃SO2.