Let $E(a,b,c;z)$ be the hypergeometric differential equation $$ z(1-z)w''+(c-(a+b+z)z)w'-abw $$ with $w$ the unknown. It is well known that if $z\notin\mathbb{Z}$, then $E(a,b,c;z)$ has linearly independent solutions around $z=0$ $$ {}_2F_1(a,b,c;z) \quad\text{and}\quad z^{1-c}{}_2F_1(1+a-c,1+b-c,2,c;z) $$ with ${}_2F_1$ the hypergeometric function.
Now, how about the case $c\in\mathbb{Z}$? Of course, we can theoretically solve the equation by using the method of Frobenius, but I would like to know explicit expressions (if there exists any explicit expression). I would also like to know the monodromies.