$\sum _{k=0}^{\infty } \sum _{v=0}^{\infty } \frac{ (a+b x)^k (a-b x)^v}{(k!)^2 v!}U(v+1,k+v+2,x)$ with $U$ being confluent hypergeometric function

Question

I am wondering if I can convert the expression

$$f(x)=\sum _{k=0}^{\infty } \sum _{v=0}^{\infty } \frac{ (a+b x)^k (a-b x)^v}{(k!)^2 v!}U(v+1,k+v+2,x)\\{\rm with}\,\,x,a\in\mathbb{R}_{>0},b\in\mathbb{R},\tag{1}$$

where $U$ is the confluent hypergeometric function, to a simpler equation. Eq.(1) is quite complicate to calculate. Possible simplifications are a reduction to a single sum or completely remove the sums, simplification of the fraction, or replacement of the hypergeometric function.