Wikipedia defines Kummer's confluent hypergeometric function $M(a,b,z) \equiv {}_1F_1(a,b,z)$ and the first solution to the Kummer's function: \begin{equation} z{\frac {d^{2}w}{dz^{2}}}+(b-z){\frac {dw}{dz}}-aw=0, \end{equation} The second solution is given to be $z^{1-b}\;{}_1F_1(a+1-b,2-b,z)$, from where we can get Tricomi's confluent hypergeometric function as the (weighted) sum of these two solutions as a solution. Thus
\begin{equation} \Psi(a,b,z) \equiv U(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a+1-b)}\ {}_1F_1(a,b,z)+\frac {\Gamma (b-1)}{\Gamma (a)}z^{1-b}\ {}_1F_1(a+1-b,2-b,z). \end{equation}
My question is, what about the weighted difference of the two solutions of the Kummer equation? Such as in:
\begin{equation} \Psi_\bullet(a,b,z) \equiv U_\bullet(a,b,z) = \frac{\Gamma(1-b)}{\Gamma(a+1-b)}\ {}_1F_1(a,b,z)-\frac {\Gamma (b-1)}{\Gamma (a)}z^{1-b}\ {}_1F_1(a+1-b,2-b,z). \end{equation}
Is that another solution? Does it have a name? Or do the weights change, and we get some other solution?
Thanks for your help here as I gain some understanding of the confluent hypergeometric function.