diff --git a/include/boost/math/special_functions/detail/hypergeometric_1F1_small_a_negative_b_by_ratio.hpp b/include/boost/math/special_functions/detail/hypergeometric_1F1_small_a_negative_b_by_ratio.hpp
index 245eb4fd4..f2814f5f8 100644
--- a/include/boost/math/special_functions/detail/hypergeometric_1F1_small_a_negative_b_by_ratio.hpp
+++ b/include/boost/math/special_functions/detail/hypergeometric_1F1_small_a_negative_b_by_ratio.hpp
@@ -45,13 +45,14 @@
          // We grab the ratio for M[a, b, z] / M[a, b+1, z] and use it to seed 2 initial values,
          // then recurse until b > 0, compute a reference value and normalize (Millers method).
          //
+         int iterations = boost::math::itrunc(-b, pol);
          boost::uintmax_t max_iter = boost::math::policies::get_max_series_iterations<Policy>();
          T ratio = boost::math::tools::function_ratio_from_forwards_recurrence(boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T>(a, b, z), boost::math::tools::epsilon<T>(), max_iter);
          boost::math::policies::check_series_iterations<T>("boost::math::hypergeometric_1F1_small_a_negative_b_by_ratio<%1%>(%1%,%1%,%1%)", max_iter, pol);
          T first = 1;
          T second = 1 / ratio;
-         int iterations = boost::math::itrunc(-b, pol);
          int scaling1 = 0;
+         BOOST_ASSERT(b + iterations != a);
          second = boost::math::tools::apply_recurrence_relation_forward(boost::math::detail::hypergeometric_1F1_recurrence_b_coefficients<T>(a, b + 1, z), iterations, first, second, &scaling1);
          int scaling2 = 0;
          first = hypergeometric_1F1_imp(a, b + iterations + 1, z, pol, scaling2);
diff --git a/include/boost/math/special_functions/hypergeometric_1F1.hpp b/include/boost/math/special_functions/hypergeometric_1F1.hpp
index 69cc13619..14ba74cac 100644
--- a/include/boost/math/special_functions/hypergeometric_1F1.hpp
+++ b/include/boost/math/special_functions/hypergeometric_1F1.hpp
@@ -256,7 +256,16 @@ namespace boost { namespace math { namespace detail {
             //
             // This is a tricky area, potentially we have no good method at all:
             //
-            if (max_b_for_1F1_small_a_negative_b_by_ratio(z) < b)
+            if (b - ceil(b) == a)
+            {
+               // Fractional parts of a and b are genuinely equal, we might as well
+               // apply Kummer's relation and get a truncated series:
+               int scaling = itrunc(z);
+               T r = exp(z - scaling) * detail::hypergeometric_1F1_imp<T>(b_minus_a, b, -z, pol, log_scaling);
+               log_scaling += scaling;
+               return r;
+            }
+            if ((b < -1) && (max_b_for_1F1_small_a_negative_b_by_ratio(z) < b))
                return hypergeometric_1F1_small_a_negative_b_by_ratio(a, b, z, pol, log_scaling);
             if ((b > -1) && (b < -0.5f))
             {
@@ -408,7 +417,6 @@ namespace boost { namespace math { namespace detail {
          // 13.3.6 appears to be the most efficient and often the most accurate method.
          return boost::math::detail::hypergeometric_1F1_AS_13_3_6(a, b, z, b_minus_a, pol, log_scaling);
       }
-
       if ((a > 0) && (b > 0) && (a * z / b > 2))
       {
          //