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 SO_n,1 transformations (principal series) can be expressed as a single integral, over a compact domain, of two matrix elements of the SO_n subgroup and a multiplier. In this way we automatically obtain their classification by the canonical chain ${\mathrm{SO}}_{\text{n, 1}}{\mathrm{\supset SO}}_n{\mathrm{\supset \cdots \supset SO}}_2$ . Analytic continuation yields the SO_n+1 matrix elements in a recursive form. We obtain the asymptotic behavior of the boost matrix elements. The Inönü‐Wigner contraction yields the ISO_n representation matrix elements classified by the chain ${\mathrm{ISO}}_n{\mathrm{\supset SO}}_n{\mathrm{\supset \cdots \supset SO}}_2$ .