I am slightly confused by the solution to the Laguerre differential equation
$$xy''+(\alpha +1-x)y' + ny=0 .$$
The solution is
$$y = c_1U(-n,1+\alpha, x) +c_2 L_n^\alpha(x),$$
where $U$ is a hypergeometric function and $L$ a Laguerre polynomial (see https://mathworld.wolfram.com/LaguerreDifferentialEquation.html). Take, for example, the special case where $n$ is a positive integer, and $\alpha=-1/2$. Then both solutions are identical. But shouldn't a second order differential equation have two linearly independent solutions?
As commented by Somos, these two solutions are correct in cases where $n$ is not an integer. Because we frequently use associated Laguerre polynomials (which are of integer order $n$), the second solution you list $U(-n, \alpha + 1, z)$ becomes proportional to $L_n^\alpha(z)$. In these cases, a second logarithmic solution must be found. Using the solutions to the confluent hypergeometric differential equation in the DLMF Chapter 13, it can be seen that these associated Laguerre solutions of the second kind can be expressed $$ \sum_{k=1}^\alpha \frac{\alpha!(k-1)!}{(\alpha-k)!(n+1)_k}z^{-k} - \sum_{k=0}^n \frac{(-n)_k}{(\alpha+1)_k k!}z^k\left[ \ln z + \psi(n+1-k)-\psi(k+1)-\psi(\alpha+k+1)\right] + \frac{\alpha!}{(n+\alpha+1)!}\frac{(-1)^{n+1}}{n+1} {_2}F_2\left(1, 1; n+2, n+\alpha+2 | z\right)$$ for nonnegative integers $n$ and $\alpha$. Here, $(x)_y$ are Pochhammer symbols, $\psi(x)$ is the digamma function, and ${_2}F_2(a_1, a_2; b_1, b_2 | x)$ is a hypergeometric function.