Hypergeometric reduction formulas involving roots of unity (Q1892551)

A generalized Kummer formula expressing Lauricella's function \(F_D^{(n- 1)}\) in variables \(\omega, \omega^2, \dots, \omega^{n-1}\), where \(\omega= \exp (2\pi i/ n)\), is derived in terms of gamma functions. If \(\{\Lambda (\mu) \}^\infty_{\mu=0}\) be a sequence of complex numbers, it is shown that the reduction formula \[ \sum_{m_1, \dots, m_n =0}^\infty \Lambda (m_1+ \cdots+ m_n) \prod_{r=1}^n {{(b)_{m_r} (x\omega^{r -1})^{m_r}} \over {m_r!}}= \sum_{j=0}^\infty {{\Lambda (nj) (b)_j x^{nj}} \over {j!}}, \] holds subject to conditions of absolute convergence. A reduction formula for a special generalized Kampé de Fériet function is derived as a special case. Further generalizations of the results are given and multiple series reduction formulae for \(F_C^{(n-1)}\) examined for \(n\leq 4\). The general case, however, is shown to be an open problem.