Considere both equations below:
\begin{align} y_{01}(z) &= H\ell(a,q;\alpha,\beta,\gamma,\delta;z)\\ y_{02}(z) &= z^{1-\gamma}H\ell(a,(a\delta+\epsilon)(1-\gamma)+q;\alpha + 1 - \gamma,\beta + 1 - \gamma,2 - \gamma,\delta;z) \end{align}
If the solutions of a Heun Equation (page 4 of the related article) around point $z = 0$ (with characteristic exponents given by $0$ and $1 - \gamma$) for the case in which $\gamma\not\in\mathbb{Z}$ are given by these equations, here's my question.
For the case in which $\gamma\in\mathbb{Z}\backslash\{1\}$, for which we have the adding of a logarithmical term on the second solution (given the fact that the difference between both the characteristic exponents is an integer now), can we write the newer two solutions (for this specific case) utilizing the format of the above equations ($y_{01}(z)$ and $y_{02}(z)$ stated at the beginning) just by properly adding the logarithmic term?