Some summation formulae for spherical spinors

Abstract

Two families of series with terms involving spherical spinors (spinor spherical harmonics) for a spin one-half particle, and are summed to closed forms involving the Legendre function of the first kind and its derivatives with respect to an argument and/or an index. In particular, it is shown that the closed forms of the series ??(0; n, n') are remarkably simple: and with I denoting the 2 ? 2 unit matrix and ? standing for the vector of Pauli matrices. Integral eigenvalue equations solved by the spherical spinors, with either ?+(?; n, n') or as kernels, are provided. It is pointed out that ?+(?; n, n') is the Green function of the operator ?? ? ? (? + 1)I, (? ? ?1, ? 2, ...), while is the generalized (reduced) Green function of the operator ?? ? ? (k + 1)I, (k = ?1, ? 2, ...), with ? = ?ir ? ?.