diff --git a/doc/source/theory/Mellin.rst b/doc/source/theory/Mellin.rst
index 2c160902f..9e4070a04 100644
--- a/doc/source/theory/Mellin.rst
+++ b/doc/source/theory/Mellin.rst
@@ -178,8 +178,14 @@ The complete list of harmonics sums available in :mod:`ekore.harmonics` is:
     - weight 5:
 
         .. math ::
-            S_{5}, S_{4,1}, S_{3,1,1}, S_{2,3}, S_{2,2,1}, S_{2,1,1,1}, S_{2,1,-2}, S_{2,-3}, S_{-2,3}, S_{-2,2,1}, S_{-2,1,1,1}, S_{-5}
-
-        these sums relies on the integrals :mod:`ekore.harmonics.f_functions` :cite:`Blumlein:2009ta`
+            S_{5}, S_{-5}
 
 We have also implemented a recursive computation of simple harmonics (single index), see :func:`ekore.harmonics.polygamma.recursive_harmonic_sum`
+
+In the extra folder we provide and additional package :mod:`harmonics_w5` which compute analytical continuation of
+weight 5 harmonics, currently not used in :mod:`ekore`:
+
+    .. math ::
+        S_{4,1}, S_{3,1,1}, S_{2,3}, S_{2,2,1}, S_{2,1,1,1}, S_{2,1,-2}, S_{2,-3}, S_{-2,3}, S_{-2,2,1}, S_{-2,1,1,1}
+
+where these sums rely on the integrals :mod:`harmonics_w5.f_functions` :cite:`Blumlein:2009ta`
diff --git a/extras/harmonics_w5/README.md b/extras/harmonics_w5/README.md
new file mode 100644
index 000000000..bed34bf91
--- /dev/null
+++ b/extras/harmonics_w5/README.md
@@ -0,0 +1,9 @@
+# Harmonics w5
+
+This is a standalone package that provides analytical continuation of weight 5 harmonics.
+
+to install it run:
+
+```
+poetry install
+```
diff --git a/extras/harmonics_w5/poetry.lock b/extras/harmonics_w5/poetry.lock
new file mode 100644
index 000000000..fec28ee43
--- /dev/null
+++ b/extras/harmonics_w5/poetry.lock
@@ -0,0 +1,1143 @@
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+]
+
+[metadata]
+lock-version = "2.0"
+python-versions = "^3.8,<3.11"
+content-hash = "f040c69cdd455f140c96b3129efcf98ff81433ae7717cbce9dbdc4ed2ed2d2f9"
diff --git a/extras/harmonics_w5/pyproject.toml b/extras/harmonics_w5/pyproject.toml
new file mode 100644
index 000000000..daa47d595
--- /dev/null
+++ b/extras/harmonics_w5/pyproject.toml
@@ -0,0 +1,90 @@
+[build-system]
+requires = ["poetry-core>=1.0.0", "poetry-dynamic-versioning"]
+build-backend = "poetry.core.masonry.api"
+
+[tool.poetry]
+name = "harmonics_w5"
+version = "0.0.0"
+description = "Weight 5 harmonics"
+readme = "README.md"
+authors = [
+  "A. Barontini <andrea.barontini@mi.infn.it>",
+  "A. Candido <alessandro.candido@mi.infn.it>",
+  "F. Hekhorn <felix.hekhorn@mi.infn.it>",
+  "N. Laurenti <niccolo.laurenti@mi.infn.it>",
+  "G. Magni <gmagni@nikhef.nl>",
+]
+classifiers = [
+  "Programming Language :: Python",
+  "Programming Language :: Python :: 3",
+  "Topic :: Scientific/Engineering :: Physics",
+  "License :: OSI Approved :: GNU General Public License v3 (GPLv3)",
+]
+repository = "https://github.com/N3PDF/eko"
+packages = [
+  { include = "harmonics_w5", from = "src" },
+]
+
+[tool.poetry.dependencies]
+python = "^3.8,<3.11"
+eko ="^0.12.0"
+
+[tool.poetry.group.test]
+optional = true
+
+[tool.poetry.group.test.dependencies]
+pytest = "^7.1.3"
+pytest-cov = "4.0.0"
+pytest-env = "^0.6.2"
+pylint = "^2.12.2"
+
+[tool.poetry.group.dev]
+optional = true
+
+[tool.poetry.group.dev.dependencies]
+pdbpp = "^0.10.3"
+ipython = "^8.0"
+
+
+[tool.poetry-dynamic-versioning]
+enable = true
+vcs = "git"
+style = "semver"
+dirty = true
+
+[tool.poetry-dynamic-versioning.substitution]
+files = ["src/harmonics_w5/version.py"]
+
+
+[tool.poe.tasks]
+test = "pytest tests"
+lint = "pylint src/**/*.py -E"
+
+[tool.pytest.ini_options]
+testpaths = ['tests/']
+python_files = ['test_*.py', 'benchmark_*.py', 'regression.py']
+python_classes = ['Test*', 'Benchmark*']
+python_functions = ['test_*', 'benchmark_*']
+env = ["D:NUMBA_DISABLE_JIT=1"]
+
+[tool.pylint.master]
+# extensions not to check
+extension-pkg-whitelist = ["numpy", "numba"]
+ignore-paths = ["tests/"]
+# has to be 1 as pylint is NOT threadsafe
+jobs = 1
+[tool.pylint.messages_control]
+disable = ["invalid-name", "fixme"]
+[tool.pylint.reports]
+# Available formats are:
+# text, parseable, colorized, json and msvs (visual studio).
+output-format = "colorized"
+[tool.pylint.format]
+# Maximum number of characters on a single line.
+max-line-length = 100
+[tool.pylint.design]
+# Maximum number of arguments for function / method
+max-args = 10
+
+[tool.pydocstyle]
+convention = "numpy"
diff --git a/extras/harmonics_w5/src/harmonics_w5/__init__.py b/extras/harmonics_w5/src/harmonics_w5/__init__.py
new file mode 100644
index 000000000..af6ccfebe
--- /dev/null
+++ b/extras/harmonics_w5/src/harmonics_w5/__init__.py
@@ -0,0 +1,387 @@
+"""Weight 5 harmonic sums."""
+
+import numba as nb
+
+from ekore.harmonics import S5, Sm5
+from ekore.harmonics.constants import log2, zeta2, zeta3, zeta4, zeta5
+from ekore.harmonics.polygamma import cern_polygamma, symmetry_factor
+
+from . import f_functions as f
+
+
+@nb.njit(cache=True)
+def S41(N, S1, S2, S3):
+    r"""Analytic continuation of harmonic sum :math:`S_{4,1}(N)`.
+
+    As implemented in eq 9.1 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    S3: complex
+        Harmonic sum :math:`S_{3}(N)`
+
+    Returns
+    -------
+    S41 : complex
+        Harmonic sum :math:`S_{4,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F9 :
+        :math:`\mathcal{M}[(\text{Li}_4(x)/(x-1))_{+}](N)`
+
+    """
+    return -f.F9(N, S1) + S1 * zeta4 - S2 * zeta3 + S3 * zeta2
+
+
+@nb.njit(cache=True)
+def S311(N, S1, S2):
+    r"""Analytic continuation of harmonic sum :math:`S_{3,1,1}(N)`.
+
+    As implemented in eq 9.21 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+
+    Returns
+    -------
+    S311 : complex
+        Harmonic sum :math:`S_{3,1,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F11 :
+        :math:`\mathcal{M}[(\text{S}_{2,2}(x)/(x-1))_{+}](N)`
+
+    """
+    return f.F11(N, S1, S2) + zeta3 * S2 - zeta4 / 4 * S1
+
+
+@nb.njit(cache=True)
+def S221(N, S1, S2, S21):
+    r"""Analytic continuation of harmonic sum :math:`S_{2,2,1}(N)`.
+
+    As implemented in eq 9.23 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    S21: complex
+        Harmonic sum :math:`S_{2,1}(N)`
+
+    Returns
+    -------
+    S221 : complex
+        Harmonic sum :math:`S_{2,2,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F11 :
+        :math:`\mathcal{M}[(\text{S}_{2,2}(x)/(x-1))_{+}](N)`
+    eko.harmonic.f_functions.F13 :
+        :math:`\mathcal{M}[(\text{Li}_{2}^2(x)/(x-1))_{+}](N)`
+
+    """
+    return (
+        -2 * f.F11(N, S1, S2)
+        + 1 / 2 * f.F13(N, S1, S2)
+        + zeta2 * S21
+        - 3 / 10 * zeta2**2 * S1
+    )
+
+
+@nb.njit(cache=True)
+def Sm221(N, S1, Sm1, S21, Sm21):
+    r"""Analytic continuation of harmonic sum :math:`S_{-2,2,1}(N)`.
+
+    As implemented in eq 9.25 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    Sm1: complex
+        Harmonic sum :math:`S_{-1}(N)`
+    S21: complex
+        Harmonic sum :math:`S_{2,1}(N)`
+    Sm21: complex
+        Harmonic sum :math:`S_{-2,1}(N)`
+
+    Returns
+    -------
+    Sm221 : complex
+        Harmonic sum :math:`S_{-2,2,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F14F12 :
+        :math:`\mathcal{M}[(2 \text{S}_{2,2}(x) - 1/2 \text{Li}_{2}^2(x))/(x+1)](N)`
+
+    """
+    return (
+        (-1) ** (N + 1) * (f.F14F12(N, S1, S21))
+        + zeta2 * Sm21
+        - 3 / 10 * zeta2**2 * Sm1
+        - 0.119102
+        + 0.0251709
+    )
+
+
+@nb.njit(cache=True)
+def S21m2(N, S1, S2, Sm1, Sm2, Sm3, S21, Sm21, S2m1):
+    r"""Analytic continuation of harmonic sum :math:`S_{2,1,-2}(N)`.
+
+    As implemented in eq 9.26 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    Sm1: complex
+        Harmonic sum :math:`S_{-1}(N)`
+    Sm2: complex
+        Harmonic sum :math:`S_{-2}(N)`
+    Sm3: complex
+        Harmonic sum :math:`S_{-3}(N)`
+    S21: complex
+        Harmonic sum :math:`S_{2,1}(N)`
+    Sm21: complex
+        Harmonic sum :math:`S_{-2,1}(N)`
+    S2m1: complex
+        Harmonic sum :math:`S_{2,-1}(N)`
+
+    Returns
+    -------
+    S21m2 : complex
+        Harmonic sum :math:`S_{2,1,-2}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F14F12 :
+        :math:`\mathcal{M}[
+            (\text{ln}(x) \text{S}_{1,2}(−x) − \text{Li}_2^2(−x)/2)/(x+1)
+            ](N)`
+
+    """
+    return (
+        (-1) ** (N) * f.F16(N, S1, Sm1, Sm2, Sm3, Sm21)
+        - 1 / 2 * zeta2 * (S21 - S2m1)
+        - (1 / 8 * zeta3 - 1 / 2 * zeta2 * log2) * (S2 - Sm2)
+        + 1 / 8 * zeta2**2 * Sm1
+        + 0.0854806
+    )
+
+
+@nb.njit(cache=True)
+def S2111(N, S1, S2, S3):
+    r"""Analytic continuation of harmonic sum :math:`S_{2,1,1,1}(N)`.
+
+    As implemented in eq 9.33 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    S3: complex
+        Harmonic sum :math:`S_{3}(N)`
+
+    Returns
+    -------
+    S2111 : complex
+        Harmonic sum :math:`S_{2,1,1,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F17 :
+        :math:`\mathcal{M}[(\text{S}_{1,3}(x)/(x-1))_{+}](N)`
+
+    """
+    return -f.F17(N, S1, S2, S3) + zeta4 * S1
+
+
+@nb.njit(cache=True)
+def Sm2111(N, S1, S2, S3, Sm1):
+    r"""Analytic continuation of harmonic sum :math:`S_{-2,1,1,1}(N)`.
+
+    As implemented in eq 9.34 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    S3: complex
+        Harmonic sum :math:`S_{3}(N)`
+    Sm1: complex
+        Harmonic sum :math:`S_{-1}(N)`
+
+    Returns
+    -------
+    Sm2111 : complex
+        Harmonic sum :math:`S_{-2,1,1,1}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F18 :
+        :math:`\mathcal{M}[\text{S}_{1,3}(x)/(x+1)](N)`
+
+    """
+    return (
+        (-1) ** (N + 1) * f.F18(N, S1, S2, S3)
+        + zeta4 * Sm1
+        - 0.706186
+        + 0.693147 * zeta4
+    )
+
+
+@nb.njit(cache=True)
+def S23(N, S1, S2, S3):
+    r"""Analytic continuation of harmonic sum :math:`S_{2,3}(N)`.
+
+    As implemented in eq 9.3 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S1: complex
+        Harmonic sum :math:`S_{1}(N)`
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    S3: complex
+        Harmonic sum :math:`S_{3}(N)`
+
+    Returns
+    -------
+    S23 : complex
+        Harmonic sum :math:`S_{2,3}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F19 :
+        :math:`\mathcal{M}[
+            ((
+                \text{ln}(x)[\text{S}_{1,2}(1-x) - \zeta_3]
+                +3 [\text{S}_{1,3}(1-x) - \zeta_4]
+            /(x-1))_{+}](N)`
+
+    """
+    return f.F19(N, S1, S2, S3) + 3 * zeta4 * S1
+
+
+@nb.njit(cache=True)
+def Sm23(N, Sm1, Sm2, Sm3, is_singlet=None):
+    r"""Analytic continuation of harmonic sum :math:`S_{-2,3}(N)`.
+
+    As implemented in eq 9.4 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    Sm1: complex
+        Harmonic sum :math:`S_{-1}(N)`
+    Sm2: complex
+        Harmonic sum :math:`S_{-2}(N)`
+    Sm3: complex
+        Harmonic sum :math:`S_{-3}(N)`
+    is_singlet: bool, None
+        symmetry factor: True for singlet like quantities (:math:`\eta=(-1)^N =
+        1`), False for non-singlet like quantities (:math:`\eta=(-1)^N=-1`)
+
+    Returns
+    -------
+    Sm23 : complex
+        Harmonic sum :math:`S_{-2,3}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F19 :
+        :math:`\mathcal{M}[
+            (
+                \text{ln}(x)[\text{S}_{1,2}(1-x) - \zeta_3]
+                +3 [\text{S}_{1,3}(1-x) - \zeta_4]
+            /(x+1)](N)`
+
+    """
+    eta = symmetry_factor(N, is_singlet)
+    return (
+        eta * f.F20(N, Sm1, Sm2, Sm3)
+        + 3 * zeta4 * Sm1
+        + 21 / 32 * zeta5
+        + 3 * zeta4 * log2
+        - 3 / 4 * zeta2 * zeta3
+    )
+
+
+@nb.njit(cache=True)
+def S2m3(N, S2, Sm1, Sm2, Sm3):
+    r"""Analytic continuation of harmonic sum :math:`S_{2,-3}(N)`.
+
+    As implemented in eq 9.5 of :cite:`Blumlein:2009ta`.
+
+    Parameters
+    ----------
+    N : complex
+        Mellin moment
+    S2: complex
+        Harmonic sum :math:`S_{2}(N)`
+    Sm1: complex
+        Harmonic sum :math:`S_{-1}(N)`
+    Sm2: complex
+        Harmonic sum :math:`S_{-2}(N)`
+    Sm3: complex
+        Harmonic sum :math:`S_{-3}(N)`
+
+    Returns
+    -------
+    S2m3 : complex
+        Harmonic sum :math:`S_{2,-3}(N)`
+
+    See Also
+    --------
+    eko.harmonic.f_functions.F19 :
+        :math:`\mathcal{M}[
+            ((
+                1/2 \text{ln}^2(x) \text{Li}_2(-x)
+                -2  \text{ln}(x) \text{Li}_3(-x)
+                +3 \text{Li}_4(-x)
+            )/(x-1)](N)`
+
+    """
+    return (
+        (-1) ** (N + 1) * f.F21(N, Sm1, Sm2, Sm3)
+        + 3 / 4 * zeta3 * (Sm2 - S2)
+        - 21 / 8 * zeta4 * Sm1
+        - 1.32056
+    )
diff --git a/src/ekore/harmonics/f_functions/__init__.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/__init__.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/__init__.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/__init__.py
diff --git a/src/ekore/harmonics/f_functions/f11.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f11.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f11.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f11.py
diff --git a/src/ekore/harmonics/f_functions/f13.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f13.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f13.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f13.py
diff --git a/src/ekore/harmonics/f_functions/f14_f12.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f14_f12.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f14_f12.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f14_f12.py
diff --git a/src/ekore/harmonics/f_functions/f16.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f16.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f16.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f16.py
diff --git a/src/ekore/harmonics/f_functions/f17.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f17.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f17.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f17.py
diff --git a/src/ekore/harmonics/f_functions/f18.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f18.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f18.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f18.py
diff --git a/src/ekore/harmonics/f_functions/f19.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f19.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f19.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f19.py
diff --git a/src/ekore/harmonics/f_functions/f20.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f20.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f20.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f20.py
diff --git a/src/ekore/harmonics/f_functions/f21.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f21.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f21.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f21.py
diff --git a/src/ekore/harmonics/f_functions/f9.py b/extras/harmonics_w5/src/harmonics_w5/f_functions/f9.py
similarity index 100%
rename from src/ekore/harmonics/f_functions/f9.py
rename to extras/harmonics_w5/src/harmonics_w5/f_functions/f9.py
diff --git a/tests/ekore/harmonics/test_f_functions.py b/extras/harmonics_w5/tests/test_f_functions.py
similarity index 98%
rename from tests/ekore/harmonics/test_f_functions.py
rename to extras/harmonics_w5/tests/test_f_functions.py
index 89136a9a6..0259adda7 100644
--- a/tests/ekore/harmonics/test_f_functions.py
+++ b/extras/harmonics_w5/tests/test_f_functions.py
@@ -1,9 +1,9 @@
 # Test F functions implementing w5 harmonics sums
 
+import harmonics_w5 as w5
 import numpy as np
 
 from ekore import harmonics
-from ekore.harmonics import w5
 
 zeta2 = harmonics.constants.zeta2
 zeta3 = harmonics.constants.zeta3
@@ -55,7 +55,7 @@
 }
 
 
-# copare the exact values of some harmonics
+# compare the exact values of some harmonics
 # All the harmonics definition are coming from :cite`:Bl_mlein_2009` section 9.
 # F19, F20,F21 are not present in that paper.
 def test_F9():
diff --git a/src/ekore/harmonics/w5.py b/src/ekore/harmonics/w5.py
index b75f9eebf..0ca6a37b8 100644
--- a/src/ekore/harmonics/w5.py
+++ b/src/ekore/harmonics/w5.py
@@ -2,14 +2,13 @@
 
 import numba as nb
 
-from . import f_functions as f
-from .constants import log2, zeta2, zeta3, zeta4, zeta5
-from .polygamma import cern_polygamma, symmetry_factor
+from .constants import zeta5
+from .polygamma import cern_polygamma
 
 
 @nb.njit(cache=True)
 def S5(N):
-    r"""Computes the harmonic sum :math:`S_5(N)`.
+    r"""Compute the harmonic sum :math:`S_5(N)`.
 
     .. math::
       S_5(N) = \sum\limits_{j=1}^N \frac 1 {j^5} = \frac 1 24 \psi_4(N+1)+\zeta(5)
@@ -72,381 +71,3 @@ def Sm5(N, hS5, is_singlet=None):
     if is_singlet:
         return 1 / 2**4 * S5(N / 2) - hS5
     return 1 / 2**4 * S5((N - 1) / 2) - hS5
-
-
-@nb.njit(cache=True)
-def S41(N, S1, S2, S3):
-    r"""Analytic continuation of harmonic sum :math:`S_{4,1}(N)`.
-
-    As implemented in eq 9.1 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    S3: complex
-        Harmonic sum :math:`S_{3}(N)`
-
-    Returns
-    -------
-    S41 : complex
-        Harmonic sum :math:`S_{4,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F9 :
-        :math:`\mathcal{M}[(\text{Li}_4(x)/(x-1))_{+}](N)`
-
-    """
-    return -f.F9(N, S1) + S1 * zeta4 - S2 * zeta3 + S3 * zeta2
-
-
-@nb.njit(cache=True)
-def S311(N, S1, S2):
-    r"""Analytic continuation of harmonic sum :math:`S_{3,1,1}(N)`.
-
-    As implemented in eq 9.21 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-
-    Returns
-    -------
-    S311 : complex
-        Harmonic sum :math:`S_{3,1,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F11 :
-        :math:`\mathcal{M}[(\text{S}_{2,2}(x)/(x-1))_{+}](N)`
-
-    """
-    return f.F11(N, S1, S2) + zeta3 * S2 - zeta4 / 4 * S1
-
-
-@nb.njit(cache=True)
-def S221(N, S1, S2, S21):
-    r"""Analytic continuation of harmonic sum :math:`S_{2,2,1}(N)`.
-
-    As implemented in eq 9.23 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    S21: complex
-        Harmonic sum :math:`S_{2,1}(N)`
-
-    Returns
-    -------
-    S221 : complex
-        Harmonic sum :math:`S_{2,2,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F11 :
-        :math:`\mathcal{M}[(\text{S}_{2,2}(x)/(x-1))_{+}](N)`
-    eko.harmonic.f_functions.F13 :
-        :math:`\mathcal{M}[(\text{Li}_{2}^2(x)/(x-1))_{+}](N)`
-
-    """
-    return (
-        -2 * f.F11(N, S1, S2)
-        + 1 / 2 * f.F13(N, S1, S2)
-        + zeta2 * S21
-        - 3 / 10 * zeta2**2 * S1
-    )
-
-
-@nb.njit(cache=True)
-def Sm221(N, S1, Sm1, S21, Sm21):
-    r"""Analytic continuation of harmonic sum :math:`S_{-2,2,1}(N)`.
-
-    As implemented in eq 9.25 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    Sm1: complex
-        Harmonic sum :math:`S_{-1}(N)`
-    S21: complex
-        Harmonic sum :math:`S_{2,1}(N)`
-    Sm21: complex
-        Harmonic sum :math:`S_{-2,1}(N)`
-
-    Returns
-    -------
-    Sm221 : complex
-        Harmonic sum :math:`S_{-2,2,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F14F12 :
-        :math:`\mathcal{M}[(2 \text{S}_{2,2}(x) - 1/2 \text{Li}_{2}^2(x))/(x+1)](N)`
-
-    """
-    return (
-        (-1) ** (N + 1) * (f.F14F12(N, S1, S21))
-        + zeta2 * Sm21
-        - 3 / 10 * zeta2**2 * Sm1
-        - 0.119102
-        + 0.0251709
-    )
-
-
-@nb.njit(cache=True)
-def S21m2(N, S1, S2, Sm1, Sm2, Sm3, S21, Sm21, S2m1):
-    r"""Analytic continuation of harmonic sum :math:`S_{2,1,-2}(N)`.
-
-    As implemented in eq 9.26 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    Sm1: complex
-        Harmonic sum :math:`S_{-1}(N)`
-    Sm2: complex
-        Harmonic sum :math:`S_{-2}(N)`
-    Sm3: complex
-        Harmonic sum :math:`S_{-3}(N)`
-    S21: complex
-        Harmonic sum :math:`S_{2,1}(N)`
-    Sm21: complex
-        Harmonic sum :math:`S_{-2,1}(N)`
-    S2m1: complex
-        Harmonic sum :math:`S_{2,-1}(N)`
-
-    Returns
-    -------
-    S21m2 : complex
-        Harmonic sum :math:`S_{2,1,-2}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F14F12 :
-        :math:`\mathcal{M}[
-            (\text{ln}(x) \text{S}_{1,2}(−x) − \text{Li}_2^2(−x)/2)/(x+1)
-            ](N)`
-
-    """
-    return (
-        (-1) ** (N) * f.F16(N, S1, Sm1, Sm2, Sm3, Sm21)
-        - 1 / 2 * zeta2 * (S21 - S2m1)
-        - (1 / 8 * zeta3 - 1 / 2 * zeta2 * log2) * (S2 - Sm2)
-        + 1 / 8 * zeta2**2 * Sm1
-        + 0.0854806
-    )
-
-
-@nb.njit(cache=True)
-def S2111(N, S1, S2, S3):
-    r"""Analytic continuation of harmonic sum :math:`S_{2,1,1,1}(N)`.
-
-    As implemented in eq 9.33 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    S3: complex
-        Harmonic sum :math:`S_{3}(N)`
-
-    Returns
-    -------
-    S2111 : complex
-        Harmonic sum :math:`S_{2,1,1,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F17 :
-        :math:`\mathcal{M}[(\text{S}_{1,3}(x)/(x-1))_{+}](N)`
-
-    """
-    return -f.F17(N, S1, S2, S3) + zeta4 * S1
-
-
-@nb.njit(cache=True)
-def Sm2111(N, S1, S2, S3, Sm1):
-    r"""Analytic continuation of harmonic sum :math:`S_{-2,1,1,1}(N)`.
-
-    As implemented in eq 9.34 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    S3: complex
-        Harmonic sum :math:`S_{3}(N)`
-    Sm1: complex
-        Harmonic sum :math:`S_{-1}(N)`
-
-    Returns
-    -------
-    Sm2111 : complex
-        Harmonic sum :math:`S_{-2,1,1,1}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F18 :
-        :math:`\mathcal{M}[\text{S}_{1,3}(x)/(x+1)](N)`
-
-    """
-    return (
-        (-1) ** (N + 1) * f.F18(N, S1, S2, S3)
-        + zeta4 * Sm1
-        - 0.706186
-        + 0.693147 * zeta4
-    )
-
-
-@nb.njit(cache=True)
-def S23(N, S1, S2, S3):
-    r"""Analytic continuation of harmonic sum :math:`S_{2,3}(N)`.
-
-    As implemented in eq 9.3 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S1: complex
-        Harmonic sum :math:`S_{1}(N)`
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    S3: complex
-        Harmonic sum :math:`S_{3}(N)`
-
-    Returns
-    -------
-    S23 : complex
-        Harmonic sum :math:`S_{2,3}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F19 :
-        :math:`\mathcal{M}[
-            ((
-                \text{ln}(x)[\text{S}_{1,2}(1-x) - \zeta_3]
-                +3 [\text{S}_{1,3}(1-x) - \zeta_4]
-            /(x-1))_{+}](N)`
-
-    """
-    return f.F19(N, S1, S2, S3) + 3 * zeta4 * S1
-
-
-@nb.njit(cache=True)
-def Sm23(N, Sm1, Sm2, Sm3, is_singlet=None):
-    r"""Analytic continuation of harmonic sum :math:`S_{-2,3}(N)`.
-
-    As implemented in eq 9.4 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    Sm1: complex
-        Harmonic sum :math:`S_{-1}(N)`
-    Sm2: complex
-        Harmonic sum :math:`S_{-2}(N)`
-    Sm3: complex
-        Harmonic sum :math:`S_{-3}(N)`
-    is_singlet: bool, None
-        symmetry factor: True for singlet like quantities (:math:`\eta=(-1)^N =
-        1`), False for non-singlet like quantities (:math:`\eta=(-1)^N=-1`)
-
-    Returns
-    -------
-    Sm23 : complex
-        Harmonic sum :math:`S_{-2,3}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F19 :
-        :math:`\mathcal{M}[
-            (
-                \text{ln}(x)[\text{S}_{1,2}(1-x) - \zeta_3]
-                +3 [\text{S}_{1,3}(1-x) - \zeta_4]
-            /(x+1)](N)`
-
-    """
-    eta = symmetry_factor(N, is_singlet)
-    return (
-        eta * f.F20(N, Sm1, Sm2, Sm3)
-        + 3 * zeta4 * Sm1
-        + 21 / 32 * zeta5
-        + 3 * zeta4 * log2
-        - 3 / 4 * zeta2 * zeta3
-    )
-
-
-@nb.njit(cache=True)
-def S2m3(N, S2, Sm1, Sm2, Sm3):
-    r"""Analytic continuation of harmonic sum :math:`S_{2,-3}(N)`.
-
-    As implemented in eq 9.5 of :cite:`Blumlein:2009ta`.
-
-    Parameters
-    ----------
-    N : complex
-        Mellin moment
-    S2: complex
-        Harmonic sum :math:`S_{2}(N)`
-    Sm1: complex
-        Harmonic sum :math:`S_{-1}(N)`
-    Sm2: complex
-        Harmonic sum :math:`S_{-2}(N)`
-    Sm3: complex
-        Harmonic sum :math:`S_{-3}(N)`
-
-    Returns
-    -------
-    S2m3 : complex
-        Harmonic sum :math:`S_{2,-3}(N)`
-
-    See Also
-    --------
-    eko.harmonic.f_functions.F19 :
-        :math:`\mathcal{M}[
-            ((
-                1/2 \text{ln}^2(x) \text{Li}_2(-x)
-                -2  \text{ln}(x) \text{Li}_3(-x)
-                +3 \text{Li}_4(-x)
-            )/(x-1)](N)`
-
-    """
-    return (
-        (-1) ** (N + 1) * f.F21(N, Sm1, Sm2, Sm3)
-        + 3 / 4 * zeta3 * (Sm2 - S2)
-        - 21 / 8 * zeta4 * Sm1
-        - 1.32056
-    )