$F(x):=\sum_{n\geq 1}\sum_{k=1}^{n} a_k\ b_{n-k}\ c_{n,k}\ x^n$.
$G(x,y,u):=\sum_{\alpha,m_1 , m_2}\ a_{\alpha}\ b_{m_2}\ c_{m_1 ,m_2}u^{\alpha}\ y^{m_2}\ x^{m_1} $
In G(x,y,z) if I put the condition that $\alpha + m_2 = m_1$ and replace $u=y$ then I should get $\sum_{n\geq 1}\sum_{k=1}^{n} a_k\ b_{n-k}\ c_{n,k}\ (ux)^n$. that is $F(xu)$.
I want to be sure of the calculation. Am I doing some mistake ?