If $u\in\mathbb{N}$ then following identify is established as an exercise in substitution $$ (-1)^{u+1}\beta\left(\frac{1}{2},u,1-u\right)=\log 2-\mathfrak{X}_{u-1}\exp(-\psi_{u-1}) \label{a}\tag{1} $$ Here $\beta$ is the incomplete beta function, $\mathfrak{X}_{u-1}$ is the numerator of the $(u-1)-$th alternating harmonic number and $\psi_{u-1}$ is the logarithm of the least common multiples of the numbers $1,2,3,\ldots,u-1.$ Indeed if we allow $u\in\mathbb{R^{+}}$ and plot the LHS of $\ref{a}$ in the Cartesian plane then we see that the zeros of the LHS of $\ref{a}$ occur whenever $u=1/2+k$ for every natural number $k.$ See the graph below. The LHS of $\ref{a}$ can be expressed as the hypergeometric function: $$ g_{u}\text{ }_{2}F_{1}\left(u,u;1+u;\frac{1}{2}\right) \label{b}\tag{2} $$ where $g_{u}:=(-1)^{1+u}(1/2)^{u}(1/u).$ It is known that $\ref{b}$ satisfies the second order linear differential equation $$ \frac{1}{2}\left(1-\frac{1}{2}\right)\frac{d^{2}w}{dz^{2}}+\left[1+u-(u+u+1)\frac{1}{2};\right]\frac{dw}{dz}-u^{2}w=0 $$ ; which after simplifying is the equation $$ \frac{d^{2}w}{dz^{2}}+2\frac{dw}{dz}-4u^2w=0 \label{c}\tag{3} $$
Question: Is equation $\ref{c}$ correct? If not what is the correct calculations?
I also think it is fair to ask what is the eigenvalue problem with respect to the Sturm Louiville form of $\ref{c}.$ Note I have the Sturm-Louiville form as: $$ \frac{dw}{dz}\left[\exp(2z)\frac{d}{dz}\right]-4\exp(2z)u^2w=0 $$
