Proof product propriety of bessel function

Question

The bessel function is given by: $${J}_{n}(x) ={ \mathop{∑ }}_{k=0}^{∞} {{(−1)}^{k}\over k!(n + k)!}{\left ({x\over 2}\right )}^{n+2k}.$$ and Translation operator can be wrighten as:
$$T_x f(y) = \frac{\Gamma(\nu+1)}{\sqrt{\pi}\Gamma\left(\nu+\frac{1}{2}\right)} \int_{0}^\pi f\left(\sqrt{x^2+ y^2-2 xy\cos(\theta) }\right)(\sin(\theta))^{2v} \,d\theta.$$ i have trouble to prove : $$T_x j_{n}(\lambda.)=j_{n}(\lambda x)j_{n}(\lambda y)$$ such that $x,y$ and $\lambda$ are $\geq 0.$