diff --git a/crates/ekore/src/harmonics/cache.rs b/crates/ekore/src/harmonics/cache.rs
index 1342f5175..9c9e3ae9b 100644
--- a/crates/ekore/src/harmonics/cache.rs
+++ b/crates/ekore/src/harmonics/cache.rs
@@ -30,6 +30,10 @@ pub enum K {
     S3mh,
     /// $g_3(N)$
     G3,
+    /// $S_{-2}(N)$ even moments
+    Sm2e,
+    /// $S_{-2}(N)$ odd moments
+    Sm2o,
 }
 
 /// Hold all cached values.
@@ -69,6 +73,8 @@ impl Cache {
             K::S2mh => w2::S2((self.n - 1.) / 2.),
             K::S3mh => w3::S3((self.n - 1.) / 2.),
             K::G3 => g_functions::g3(self.n, self.get(K::S1)),
+            K::Sm2e => w2::Sm2e(self.get(K::S2), self.get(K::S2h)),
+            K::Sm2o => w2::Sm2o(self.get(K::S2), self.get(K::S2mh)),
         };
         // insert
         self.m.insert(k, val);
diff --git a/crates/ekore/src/harmonics/w2.rs b/crates/ekore/src/harmonics/w2.rs
index d55a32679..a6d7cc89f 100644
--- a/crates/ekore/src/harmonics/w2.rs
+++ b/crates/ekore/src/harmonics/w2.rs
@@ -11,3 +11,13 @@ use crate::harmonics::polygamma::cern_polygamma;
 pub fn S2(N: Complex<f64>) -> Complex<f64> {
     -cern_polygamma(N + 1.0, 1) + ZETA2
 }
+
+/// Analytic continuation of harmonic sum $S_{-2}(N)$ for even moments.
+pub fn Sm2e(hS2: Complex<f64>, hS2h: Complex<f64>) -> Complex<f64> {
+    1. / 2. * hS2h - hS2
+}
+
+/// Analytic continuation of harmonic sum $S_{-2}(N)$ for odd moments.
+pub fn Sm2o(hS2: Complex<f64>, hS2mh: Complex<f64>) -> Complex<f64> {
+    1. / 2. * hS2mh - hS2
+}