acb_hypgeom_m_asymp, acb_hypgeom_m_1f1, acb_hypgeom_m, acb_hypgeom_1f1

acb_hypgeom_m_asymp uses the asymptotic expansions of U^{*} through the connection formula

acb_hypgeom_m_1f1 sums the convergent series, after Kummer's transformation M(a,b,z) = e^z M(b-a,b,-z) if \operatorname{Re} z < 0 (unless the series terminates), and handles \mathbf{M} at nonpositive integer b by the limit formula. acb_hypgeom_m chooses: terminating series are summed directly; otherwise it compares the accuracy of the asymptotic expansion (about |z|/\log 2 bits) with that of the convergent series after cancellation, uses Kummer's transformation for negative real part, and sets the working precision accordingly. If the result has less than half the requested accuracy and all arguments are real and moderate, it also tries arb_hypgeom_1f1_integration and keeps the better enclosure. See Confluent hypergeometric functions.