Factorized representation for parity-projected Wigner ${\mathrm{d}}^j\left(\beta \right)$ matrices

An alternative representation for the parity-projected Wigner ${\mathbf{d}}^j\left(\beta \right)$ rotation matrix is derived as the product of two triangular matrices composed of Gegenbauer polynomials with negative and positive upper indices, respectively. We relate this representation for ${\mathbf{d}}^j\left(\beta \right)$ to the one presented by Matveenko [Phys. Rev. A 59, 1034 (1999)], which, in contrast with our result, requires for its evaluation a matrix inversion. In addition, identities for bilinear sums of Gegenbauer polynomials are derived. This work is based on our recently introduced invariant representations for finite rotation matrices [Phys. Rev. A 57, 3233 (1998)].