Expression of special stretched $9j$ coefficients in terms of $_5F_4$ hypergeometric series

TL;DR

This work investigates whether special stretched $9j$ coefficients can be written as a $_5F_4(1)$ hypergeometric series. By leveraging the Dougall-Ramanujan identity for a well-poised $_5F_4(1)$ and a targeted change of variables, the authors derive an explicit $_5F_4(1)$ representation for a doubly stretched $9j$ with two degenerate triads, yielding a compact, exact expression up to a prefactor. They compare this form to existing Racah-like representations and demonstrate significant computational efficiency gains when evaluating the hypergeometric form. The study concludes that while a general $9j$ cannot be captured by $_5F_4(1)$, this specific nontrivial case reveals a promising pathway for efficient angular-momentum coefficient calculations and motivates search for additional special cases.

Abstract

The Clebsch-Gordan coefficients or Wigner $3j$ symbols are known to be proportional to a $_3F_2(1)$ hypergeometric series, and Racah $6j$ coefficients to a $_4F_3(1)$. In general, however, non-trivial $9j$ symbols can not be expressed as a $_5F_4$. In this letter, we show, using the Dougall-Ramanujan identity, that special stretched $9j$ symbols can be reformulated as $_5F_4(1)$ hypergeometric series.