Good day to everyone. I've got in a pickle while toying around with some transformations.
It is well-known that the bivariate confluent hypergeometric function $\Phi_2(\cdot)$ can be expanded in the following way: $$\Phi_2\left(b,b';c;w,z\right)=\sum_{l=0}^{\infty}\frac{(b)_l}{(c)_l}\frac{w^l}{l!}\mbox{}_1F_1(b';c+l;z),$$ where $\mbox{}_1F_1(b';c+l;z)$ is the Kummer confluent hypergeometric function.
The question is whether it is possible to define a function given by the similar series, but with a Tricomi confluent hypergeometric function ($U(b';c+1+l;z)$), i.e., $$\sum_{l=0}^{\infty}\frac{(b)_l}{(c)_l}\frac{w^l}{l!}U(b';c+1+l;z)=?.$$
Seems odd, but up to now I did not manage to convert it to any form of analytic functions.