P-symbols, Heun identities, and 3F2 identities

The usefulness of Riemann P-symbols in deriving identities involving the parametrized special function Hl is explored. Hl is the analytic local solution of the Heun equation, the canonical second-order differential equation on the Riemann sphere with four regular singular points. The identities discussed include ones coming from Mobius automorphisms and F-homotopies. and also quadratic and biquadratic transformations. The case when Hl is identical to a generalized hypergeometric function of F-3(2) type is examined. and Pfaff and Euler transformations of F-3(2) (a(1), a(2), e + 1; b(1), e; x) are derived. They extend several F-3(2) identities of Bailey and Slater.