How does one find the solution of
$$\dfrac{dy}{dx}\left( 1-\left( 1-t\right) x-x^{2}\right) -\left( 1+h\left( 1+t\right) +x\right) y=0\quad ?$$
where $h$ is an integer constant and $t$ is a real constant between $0$ and $1$.
$($ In Roger Apéry, Interpolations de Fractions Continues et Irrationalité de certaines Constantes, Bull. section des sciences du C.T.H.S., n.º3, p.37-53, the solution is
$$y=(1-x)^{-1-h}(1+tx)^{h}.)$$
Note: The sequence $(v_{h,n})$ in $y=f_{h}(x)=\displaystyle\sum_{n\ge 0}v_{h,n}x^n$ satisfies a recurrence related to $\log (1+t)$.
Added: Copy of the original with the equation and solution
Addendum 2: I transcribe the comment in the 1st answer: "the corrected differential equation above agrees with the recurrence in your excerpt so there is clearly a typo in the printed differential equation."