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How can we code the hypergeometric function $$ f(n,k,z)={_nF}_{n-1}{\huge(}{1-k,\overbrace{2,\dots,2}^{n-1\ \text{times}} \atop \underbrace{1,\dots,1}_{n-1\ \text{times}}};z{\huge)} $$ as a function that accepts any $n=1,2,3,\dots$? This function reduces to a rational function in $z$ but that is not what I'm after. I apologize for lack of an attempt here...did not have any idea where to start.

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I would do it this way:

aaronF[n_Integer?Positive, k_, z_] :=
  HypergeometricPFQ[Prepend[ConstantArray[2, n - 1], 1 - k], ConstantArray[1, n - 1], z]

For example:

aaronF[5, 4, z]
   1 - 48 z + 243 z^2 - 256 z^3
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  • $\begingroup$ @JM Thank you. Exactly what I was looking for. $\endgroup$ – Aaron Hendrickson May 30 at 13:17

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