I want to compute the square modulus of the following sum :
\begin{align} \sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x) \end{align}
Where p is an integer, j is an integer, eta is a real constant, lambda is a complex number of modulus 1, and $J_{j-p}(x)$ is the Bessel function of the first kind of order (j-p).
I know from here that this should be equal to :
\begin{align} \vert \sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x) \vert^2 = J^2_{j}(x) \end{align}
To prove this, I start as following :
\begin{align} \vert \sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x) \vert^2 = \vert \left(\sum_p e^{i\eta p}(-\lambda)^{j-p} J_{j-p}(x) \right)^2\vert = \vert \sum_p\sum_q e^{i\eta p+q}(-\lambda)^{2j-p+q} J_{j-p}(x) J_{j-q}(x)\vert \end{align}
But I am stuck : which Bessel function properties I should use to arrive at the result found in the aforementioned paper? At some point I imagine orthogonality of Bessel functions is involved, but I am not sure where to start.
Does someone has a way to prove this?