Hypergeometric $\lim_{z\to 1}{}_3F_2(a,b,c;d,e;z)$ with $a+b+c-d-e=0$?

Question

Is there anything special one can say about a hypergeometric function

$$\lim_{z\to 1}{}_3F_2\left({{a,b,c}\atop{d,e}};z\right)$$

in the case when $a+b+c-d-e=0$?

Answer 1

If $a+b+c=d+e$ then ${_3F}_2(\mathbf a;\mathbf b;z)$ is said to be zero-balanced which results in a logarithmic singularity at $z=1$. Specifically, as $z\to 1$: $$ {_3F}_2\left({a,b,c \atop d,e};z\right)\sim-\frac{\Gamma(d)\Gamma(e)}{\Gamma(a)\Gamma(b)\Gamma(c)}\log(1-z). $$ Hence $$ \lim_{z\to 1}{_3F}_2\left({a,b,c \atop d,e};z\right)=\infty. $$

See here for more details.