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I try to calculate

    DifferenceRootReduce[(
 Sqrt[\[Pi]]
   Gamma[3/2 + k] HypergeometricPFQ[{3/2 + k, -n}, {2 + k}, p])/
 Gamma[2 + k], k] 

but any result but if using functionExpand to get

(Sqrt[\[Pi]]
   Gamma[3/2 + k] HypergeometricPFQ[{3/2 + k, -n}, {2 + k}, p])/
 Gamma[2 + k] == (
 Sqrt[\[Pi]] Gamma[3/2 + k] Hypergeometric2F1[3/2 + k, -n, 2 + k, p])/
 Gamma[2 + k]

and calculating the same function as Hypergeometric2F1 I get the result as following

DifferenceRootReduce[(
 Sqrt[\[Pi]] Gamma[3/2 + k] Hypergeometric2F1[3/2 + k, -n, 2 + k, p])/
 Gamma[2 + k], k] 

and the get the result

DifferenceRoot[
  Function[{\[FormalY], \[FormalN]}, {(3 + 
         2 \[FormalN]) Sqrt[\[Pi]] \[FormalY][\[FormalN]] - (4 + 
         2 \[FormalN] + 5 p + 2 \[FormalN] p + 
         2 n p) Sqrt[\[Pi]] \[FormalY][1 + \[FormalN]] + 
      2 (3 + \[FormalN] + n) p Sqrt[\[Pi]] \[FormalY][
        2 + \[FormalN]] == 0, \[FormalY][0] == 
     1/2 \[Pi] Hypergeometric2F1[3/2, -n, 2, p], \[FormalY][1] == 
     3/8 \[Pi] Hypergeometric2F1[5/2, -n, 3, p]}]][k]

maybe it is a error of my 11.1 version of Mathematica and it is posible avoid it thanks anyway

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