See the enclosed file.
In https://en.wikipedia.org/wiki/Hypergeometric\_function
we can find
The hypergeometric function is a solution of Euler's hypergeometric differential equation
$$ (1) H_{abc}w=z(1-z){\frac {d^{2}w}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac {dw}{dz}}-ab\,w=0. $$
which has three regular singular points: $0,1$ and $\infty$.
We will write simple $F$ instead of $_{2}F_{1}$.
\bques
What are solution of Euler's hypergeometric differential equation $H_{abc}w==0$ with singular point at $1$.
\eques
Around $z = 1$, if $c- a - b$ is not an integer, one has two independent solutions
${\displaystyle \,_{2}F_{1}(a,b;1+a+b-c;1-z)}$
and
${\displaystyle (1-z)^{c-a-b}\;_{2}F_{1}(c-a,c-b;1+c-a-b;1-z)}$.
What are solutions of (1) if $c- a - b$ is an integer and what is known in the literature ?
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