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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 $a+b+c-d-e=0$?

Kagaratsch
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  • I used Mathematica ''FullSimplify[HypergeometricPFQ[{a, b, c}, {d, e}, 1], Assumptions -> Element[a | b | c | d | e, Integers] && a + b + c - d - e == 0]'', and there is no simplification. – Aymane Fihadi Mar 16 '17 at 21:37
  • What I notice is that trying to evaluate it at various $a,b,c,d,e$ (not integers) seems to give infinity. Maybe I should not put $z=1$ but rather consider the limit $z\to 1$. I'll update the question. – Kagaratsch Mar 16 '17 at 21:41

1 Answers1

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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.

Aaron Hendrickson
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