I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch Gordan coefficients.

I would be very grateful if you could share with me a closed form of the ratio between

$$ \begin{pmatrix} l_1 &l_2 &l_3\\ 0&2&-2 \end{pmatrix} $$

and

$$ \begin{pmatrix} l_1 &l_2 &l_3\\ 0&0&0 \end{pmatrix} \;, $$

for $l_1+l_2+l_3$ even, if it exists. Ideally, I would like an expansion of the first 3j-symbol in terms of the second one, i.e.

first_3j = second_3j * ( 1 + ... )

Thank you for your consideration.

Best wishes, Guido

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan coefficients.

I would be very grateful if you could share with me a closed form of the ratio between

$$ \begin{pmatrix} l_1 &l_2 &l_3\\ 0&2&-2 \end{pmatrix} $$

and

$$ \begin{pmatrix} l_1 &l_2 &l_3\\ 0&0&0 \end{pmatrix} \;, $$

for $l_1+l_2+l_3$ even, if it exists. Ideally, I would like an expansion of the first 3j-symbol in terms of the second one, i.e.

first_3j = second_3j * ( 1 + ... )

Thank you for your consideration.

Best wishes, Guido