Summation Theorems for Hypergeometric Series in $U(n)$

Abstract

Explicit expressions for the matrix elements of multiplicity-free Wigner and Racah coefficients in $U(n)$ are used to establish multidimensional analogs of known hypergeometric summation theorems. Infinite sequences of such analogs are found for the Gauss theorem, Saalschütz’ theorem, and the summation theorem for well-poised ${}_5 F_4 (1)$. For the sake of completeness, results published earlier are also presented: an infinite sequence of summation theorems analogous to that for well-poised ${}_4 F_3 (-1)$ and a single analog to Whipple's theorem which arises from the construction of tensor operators in $U(3)$.