diff --git a/crates/ekore/src/harmonics/cache.rs b/crates/ekore/src/harmonics/cache.rs
index da0f4800b..f8dfd487c 100644
--- a/crates/ekore/src/harmonics/cache.rs
+++ b/crates/ekore/src/harmonics/cache.rs
@@ -46,6 +46,10 @@ pub enum K {
     Sm21e,
     /// $S_{-2,1}(N)$ odd moments
     Sm21o,
+    /// recursive harmonics
+    S1ph,
+    S2ph,
+    S3ph,
 }
 
 /// Hold all cached values.
@@ -98,6 +102,9 @@ impl Cache {
             K::Sm3o => w3::Sm3o(self.get(K::S3), self.get(K::S3mh)),
             K::Sm21e => w3::Sm21e(self.n, self.get(K::S1), self.get(K::Sm1e)),
             K::Sm21o => w3::Sm21o(self.n, self.get(K::S1), self.get(K::Sm1o)),
+            K::S1ph => recursive_harmonic_sum(self.get(K::S1mh), (self.n - 1.) / 2., 1, 1),
+            K::S2ph => recursive_harmonic_sum(self.get(K::S2mh), (self.n - 1.) / 2., 1, 2),
+            K::S3ph => recursive_harmonic_sum(self.get(K::S3mh), (self.n - 1.) / 2., 1, 3),
         };
         // insert
         self.m.insert(k, val);