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I just tried in Mathematica 11.3

SeriesCoefficient[Hypergeometric2F1[1, 1 - n, 2 + n, w], {w, 0, m}, Assumptions -> n > m > 1 && n \[Element] Integers]

which gives $$ \frac{(n+1)! (m-n)!}{(-n)! (m+n+1)!} $$

This cannot be correct, since $m-n<0$ and its factorial is infinity. Is there any walkaround for this problem?

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  • $\begingroup$ You have negative factorials in the numerator and denominator, so the ratio is regular and equal to $\frac{(-1)^m (n-1)! (n+1)!}{(n-m-1)! (n+m+1)!}$ for $n > m > 1$ and $n$,$m$ integers. $\endgroup$ – QuantumDot Jan 13 at 15:25
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I'd say it was a bug. Here's a better expression:

FunctionExpand@
  SeriesCoefficient[Hypergeometric2F1[1, 1 - n, 2 + n, w], {w, 0, m}, 
   Assumptions -> n > m > 1 && n ∈ Integers] //. {Gamma[a_]/
   Gamma[b_] :> Pochhammer[b, a - b]}
(*  Pochhammer[1 - n, m] Pochhammer[2 + m + n, -m]  *)
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