diff --git a/math.jl b/math.jl
index 96b6e7d..fb55ec9 100644
--- a/math.jl
+++ b/math.jl
@@ -1,4 +1,4 @@
-using SpecialFunctions
+using SpecialFunctions, WignerSymbols
 include("helper.jl")
 
 # Gaussian potentials in HO space
@@ -13,3 +13,26 @@ sqrt_double_factorial(n) = Iterators.prod(sqrt.(n:-2:1))
 sqrt_sqrt_pi = sqrt(sqrt(pi))
 laguerre(l, n, x) = gamma(n + l + 3/2) * better_sum([(-x * x)^k / gamma(k + l + 3/2) * inv_factorial(n - k) * inv_factorial(k)  for k in 0:n])
 ho_basis(l, n, x) = (-1)^n / sqrt_sqrt_pi * 2^((n + l + 2) / 2) * sqrt_factorial(n) / sqrt_double_factorial(2*n + 2*l + 1) * x^(l + 1) * exp(-x^2 / 2) * laguerre(l, n, x)
+
+# for implementation of simple relative coordinates
+double_factorial(n) = Iterators.prod(n:-2:1)
+
+"Gaussian integral for n ∈ Integers (Ref: Wolfram MathWorld + simplifications)"
+gauss_int(a, n) = double_factorial(n - 1) / (2 * a)^((n + 1)/2) * (iseven(n) ? sqrt(π / 2) : 1)
+
+"Gives ∫dp p u' u where u' and u' may have different l (Ref: worked out in Mathematica)"
+function integral(np, lp, n, l, μω)
+    s = 0.0
+    for m in 0 : n
+        for mp in 0 : np
+            s += (-1)^(m + mp) * gauss_int(1, 2 * m + 2 * mp + l + lp + 3) / (factorial(n - m) * gamma(l + 3/2 + m) * factorial(m) * factorial(np - mp) * gamma(lp + 3/2 + mp) * factorial(mp))
+        end
+    end
+    return 2 * sqrt(μω * factorial(n) * factorial(np) * gamma(n + l + 3/2) * gamma(np + lp + 3/2)) * s
+end
+
+"Gives <n' l'|| p ||n l> for the HO basis"
+reduced_matrix_element(np, lp, n, l, μω) = (-1)^lp * sqrt(2*lp + 1) * sqrt(2*l + 1) * wigner3j(Float64, lp, 1, l, 0, 0, 0) * integral(np, lp, n, l, μω)
+
+"Matrix element <n1p l1p n2p l2p| p1⋅p2 |n1 l1 n2 l2> (Ref: de-Shalit & Talmi, Eq 15.5)"
+racahs_reduction_formula(n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2) = (-1)^(l1 + l2p + Λ) * wigner6j(Float64, l1p, l2p, Λ, l2, l1, 1) * reduced_matrix_element(n1p, l1p, n1, l1, μ1ω1) * reduced_matrix_element(n2p, l2p, n2, l2, μ2ω2)
diff --git a/test/racah_test_data.csv b/test/racah_test_data.csv
new file mode 100644
index 0000000..657c6d4
--- /dev/null
+++ b/test/racah_test_data.csv
@@ -0,0 +1,394 @@
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+8,2,6,1,2,0,10,3,3,1.794586611044986,2.4891313351625364,0.
+2,2,9,1,7,2,7,3,3,1.6950266216464565,1.9267314398803421,0.
+8,2,2,3,5,2,8,2,1,1.240660729352081,0.614397197033274,0.
+10,2,7,0,2,3,1,1,2,0.942161064732737,1.7768239062076931,0.
+8,2,2,2,9,2,3,1,3,0.8801070256690937,1.3334106960769039,0.
+0,1,3,0,7,0,8,1,1,2.759575651836407,2.2370044561599594,0.
+2,0,2,1,1,2,9,2,1,1.0254183581144791,2.38137172597511,0.
+2,2,4,0,10,2,8,3,2,1.8112137495609861,2.6613449708817436,0.
+7,1,6,2,4,2,9,1,1,1.5940820518066428,1.0983546198249372,0.
+8,1,9,0,1,2,1,3,1,0.8932685422132205,2.9473230283991008,0.
+9,2,2,2,3,0,1,1,1,0.33570537722771965,2.0374359188831273,0.
+9,2,7,1,3,3,2,1,2,0.8073783504615606,1.9841095317741253,0.
+2,3,3,1,9,3,1,2,2,2.8250883094896393,2.258440952824621,0.
+7,0,1,3,5,1,6,3,3,1.287902963611098,2.3433999040190434,0.
+7,0,5,2,9,1,7,2,2,1.1234261485960038,0.3136829770445666,0.
+9,3,8,0,7,3,10,3,3,1.0048701834268332,2.6402255736028826,0.
+6,0,2,2,4,3,7,1,2,1.759757357100633,2.2623048177547833,0.
+4,0,1,1,3,1,4,2,1,2.251256964944865,1.776575907658561,0.
+4,0,9,2,8,1,2,2,2,1.936537095453656,1.9097068100347103,0.
+4,2,10,0,5,1,3,3,2,0.8590177771821113,1.8394463532255436,0.
+0,0,3,0,10,2,2,2,0,2.8640285895042297,0.8791871038094534,0.
+9,1,3,3,8,1,9,2,2,1.1834517327814233,2.8239738427124843,0.
+2,3,5,0,4,0,0,3,3,0.6805254666938301,0.26557352050621175,0.
+5,3,8,2,3,0,10,1,1,1.3910977429097215,1.8741960916665903,0.
+7,2,6,3,2,3,7,0,3,1.3872547516547633,1.6963007128362868,0.
+6,1,3,2,6,3,4,3,1,1.0626034805404814,2.8060268100430026,0.
+5,3,1,3,3,0,4,3,3,0.7430544120091587,1.0777388348239691,0.
+5,3,1,2,5,3,1,3,2,1.3542358203080034,0.26106660478213595,0.
+4,2,6,1,7,1,1,3,3,2.820371318333427,2.9207841371657253,0.
+8,2,5,3,1,2,5,1,1,0.27886042485737006,2.4622975712253945,0.
+1,3,9,3,7,1,4,2,1,0.3062576036365292,0.947350388684844,0.
+10,1,9,0,1,1,1,2,1,2.842529772037765,1.72598770316766,0.
+3,3,0,0,7,0,0,3,3,2.9419931449815655,1.5415649267198361,0.
+7,1,7,1,1,3,2,1,2,0.7322040099239033,1.4350899704211022,0.
+8,3,1,2,10,2,10,1,3,1.1102251621724681,1.3819727089193692,0.
+4,2,8,0,10,2,4,1,2,0.8009382774044647,0.3029995840779338,0.
diff --git a/test/racahs_reduction_formula.jl b/test/racahs_reduction_formula.jl
new file mode 100644
index 0000000..344a9ee
--- /dev/null
+++ b/test/racahs_reduction_formula.jl
@@ -0,0 +1,39 @@
+using QuadGK
+include("../math.jl")
+
+# Testing ∫dp p u' u where u' and u' may have different l
+
+function explicit_integral(np, lp, n, l, μω)
+    integrand(p) = p * (-1)^(n + np) * 1/sqrt(μω) * ho_basis(l, n, 1/sqrt(μω) * p) * ho_basis(lp, np, 1/sqrt(μω) * p)
+    (integral, _) = quadgk(integrand, 0, Inf; atol=1e-10)
+    return integral
+end
+
+n_list = 0 : 5
+l_list = 0 : 5
+μω_list = [1, 2.66]
+
+for np in n_list
+    for n in n_list
+        for lp in l_list
+            for l in l_list
+                for μω in μω_list
+                    println("Checking n=$n, np=$np, l=$l, lp=$lp, μω=$μω")
+                    expl = explicit_integral(np, lp, n, l, μω)
+                    anal = integral(np, lp, n, l, μω)
+                    @assert isapprox(expl, anal; atol=1e-8) "Explicit=$(expl) and analytical=$(anal)"
+                end
+            end
+        end
+    end
+end
+
+# Testing the full Racah's reduction formula
+
+test_data = DelimitedFiles.readdlm("test/racah_test_data.csv", ',') # format = n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2, racah_matrix_element
+
+for i in 1 : size(test_data)[1]
+    n1p, l1p, n2p, l2p, n1, l1, n2, l2, Λ, μ1ω1, μ2ω2, racah_matrix_element = test_data[i, :]
+    val = racahs_reduction_formula(Int(n1p), Int(l1p), Int(n2p), Int(l2p), Int(n1), Int(l1), Int(n2), Int(l2), Int(Λ), μ1ω1, μ2ω2)
+    @assert isapprox(val, racah_matrix_element; atol=1e-8) "Formula=$(val) and test reference=$racah_matrix_element"
+end
\ No newline at end of file