Applications of quadratic and cubic hypergeometric transformations (Q6595840)

The authors consider five classes of quadratic and cubic hypergeometric transformations and evaluate hypergeometric functions of the type \({}_2 F_1(x)\), \({}_3 F_2(x)\), and \({}_4F_3(x)\), where \(x\in \{-8, -\frac18, \pm 1, \frac14 \}\). For each case, companion identities and/or inverse transformations are derived, which sometimes can be proved by a limiting process for a divergent hypergeometric series. Several proofs use the Clausen quadratic formula, the Euler reflection formula, Legendre duplication, the Gauss multiplication formula, the Euler transformation, the hypergeometric reversion formula, and other known hypergeometric summation formulas. The proofs in the terminating case turn out to be simpler and lead to mixed summation formulas which depend on values of a certain negative integer.