Does the following identity hold?
$$\sum_{k= 0}^\infty\sum_{l=0}^\infty\frac{\alpha^{2k+l}(\lambda+\nu)_{2k+l}}{4^kl!k!(\nu+1)_k(\lambda+\nu+\mu)_{2k+l}} = {}_2F_2\left(\nu+\frac{1}{2},\lambda+\nu;2\nu+1,\mu+\lambda+\nu;2\alpha\right),$$where $\lambda,\mu,\nu>0$, and ${}_2F_2$ is the generalized hypergeometric function, and $(a)_n=a(a+1)\cdots (a+n-1)$ is the Pochhammer symbol? I found that this identity needs to hold as an intermediate step while solving an integration, which can be found as the $4$th integral in page $703$ of the book by Gradshteyn and Ryzhik.
The main problem I face is the double summation in the left hand side of the identity. How should one proceed to convert this into a single summation to derive the hypergeometric function? Any ideas? Please help. Thanks in advance.