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my dear fellows,

I have a question to make. Given the hypergeometric function $_{2}F_{1}[a,b,c,z]$ in the interval $z \in (1, \infty)$. What is the proper asymptotic expansion of the aforesaid function near $z=1$, when one is approaching from $z>1$.

Any help that anyone can kindly provide will be much appreciated.

M.

user630770
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    which $a,b,c$ are you considering ? or what is the relation among them? – G Cab Feb 17 '19 at 13:58
  • Hello, I am considering the case where c=a+b. I am trying to find the asymptotic form of the hypergeometric near z=1 but for z \in (1, +\infty). Any ideas? – user630770 Feb 18 '19 at 20:43
  • did you take a look here ? – G Cab Feb 18 '19 at 22:25
  • Hi G Cab, I looked there but still it does not have an asymptotic expansion at $z=1$ when $z \geq 1$. Thanks anyway!! – user630770 Feb 19 '19 at 10:12
  • Please describe what you tried and what's your level of knowledge of HG. Mine is medium and mainly applicative. – G Cab Feb 21 '19 at 22:32
  • $c = a + b$ is a logarithmic case, where $${_2\hspace{-1px}F_1}(a, b; a + b; z) \sim -\frac {\Gamma(a + b)} {\Gamma(a) \Gamma(b)} \ln(1 - z), \quad z \to 1^-.$$ The values for $z > 1$ will depend on how you choose the analytic continuation. You can define it in such a way that the leading term is the same as above for $z \to 1^+$. – Maxim Feb 22 '19 at 17:23
  • Thanks Maxim for the help. M. – user630770 Feb 23 '19 at 00:22

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