Let $K = SU(2) = \{ k[\alpha ,\beta] | \alpha ,\beta \in \mathbb{C}, |\alpha |^2 + |\beta |^2 = 1 \} $ with 
$$ k [ \alpha , \beta ] = 
\begin{pmatrix}
	\alpha & \beta \\
	- \overline{\beta} & \overline{\alpha}
\end{pmatrix}$$
Let $V_n$ denote the space of polynomials in one variable of degree at most $n$.  We choose the set $\{ z^{l-q} : q\in\mathbb{Z},\  |q|\leq l\}$ as a basis for $V_{2l}$. Then the (irreducible and unitary) representations of $K$ are given by the formula
$$ \sigma_l(k [\alpha , \beta]) z^{l-q} = ( \alpha z - \overline{\beta} )^{l-q} (  \beta  z + \overline{\alpha})^{l+q} .$$
Let $ \Phi_{p,q}^l $ denote the coefficient of $z^{l-p}$ in the polynomial expansion of $\sigma_l(k [\alpha , \beta]) z^{l-q}$. i.e,
$$ \sum_{|p|\leq l} \Phi_{p,q}^l(k[\alpha,\beta]) z^{l-p} = ( \alpha z - \overline{\beta} )^{l-q} (  \beta  z + \overline{\alpha})^{l+q} ,\quad for \ |q|\leq l. $$

There is an explicit formula of
$$ \int_K \Phi_{p,q}^l(k) \overline{\Phi_{p_1,q_1}^{l_1}(k)} dk$$
in the book 'Sum Formula for SL2 over Imaginary Quadratic Number Fields' by Lokvenec-Guleska. 
(https://dokumen.tips/documents/sum-formula-for-sl2-over-imaginary-quadratic-number-fields-za-sumirae-za-sl-2.html?page=23)

I have to solve the triple product
$$ \int_K \Phi_{p,q}^l(k) \Phi_{p_1,q_1}^{l_1}(k) \Phi_{p_2,q_2}^{l_2}(k) dk.$$

I know how to compute this equation through brute force. However, my advisor doesn’t want me to include the complex calculation process in my paper. Is there a place where I can directly find this formula? Any suggestion would be helpful. Thanks!

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In fact I have obtained the following formula: For any $a,b,c,d \in\mathbb{Z}$ and $k = k[\alpha,\beta]$, we have
	$$\int_K \alpha^{a}  \bar{\alpha} ^{b} \beta^{c}  \bar{\beta} ^{d} d k = \delta ( a=b, c=d ) \frac{a!c!}{(a+c+1)!},$$
where $\delta$ denotes the Kronecker symbol.