Needs["SpecialFunctions`"]

F = ((Binomial[n, k] (x^k) (y^(n - k)))/(((x + y))^n))

R = ((k y)/((k - n - 1) (x + y)))

ratk = SimplifyCombinatorial[((F /. k -> k + 1)/F)]

ratn = SimplifyCombinatorial[((F /. n -> n + 1)/F)]

Together[
      ratn - 1 - ((R /. k -> k + 1) ratk - R)]

(∑_(k = 0)^0) F /. n -> 0

F = ((k Binomial[n, k])/(n (2^(n - 1))))

R = ((k - 1)/(2 (k - n - 1)))

ratk = SimplifyCombinatorial[((F /. k -> k + 1)/F)]

ratn = SimplifyCombinatorial[((F /. n -> n + 1)/F)]

Together[
      ratn - 1 - ((R /. k -> k + 1) ratk - R)]

(∑_(k = 0)^1) F /. n -> 1

\(F = Binomial[n, k]\^2\/Binomial[2  n, n]\)

R = (((k^2) (2 k - 3 n - 3))/(2 (((k - n - 1))^2) (2 n + 1)))

ratk = SimplifyCombinatorial[((F /. k -> k + 1)/F)]

ratn = SimplifyCombinatorial[((F /. n -> n + 1)/F)]

Together[
      ratn - 1 - ((R /. k -> k + 1) ratk - R)]

(∑_(k = 0)^0) F /. n -> 0

F = 
      HyperTerm[{-n, a}, {b}, 1, k]/(Pochhammer[b - a, n]/Pochhammer[b, n])

R = ((k (b + k - 1))/((k - n - 1) (-a + b + n)))

ratk = SimplifyCombinatorial[((F /. k -> k + 1)/F)]

ratn = SimplifyCombinatorial[((F /. n -> n + 1)/F)]

Together[
      ratn - 1 - ((R /. k -> k + 1) ratk - R)]

(∑_(k = 0)^0) F /. n -> 0

Clear[WZCertificate]
WZCertificate[F_, k_, n_] := 
      Module[{a, 
          G}, 
a = (F /. n -> n + 1) - 
            F; 
G = 
          DiskreteStammFunktion[a, 
            k]; 
SimplifyCombinatorial[
          G/F]
]

Clear[WZCertificate]
\(WZCertificate[F_, k_, n_] := 
      Module[{ratk, ratn, rat, u, v, M, dis, h, j, A, B, CC, gcd, g}, 
        ratk = SimplifyCombinatorial[\((F /. k -> k + 1)\)/
              F]; 
ratn = 
          SimplifyCombinatorial[\((F /. n -> n + 1)\)/
              F]; 
rat = 
          SimplifyCombinatorial[
            ratk*\((\((ratn /. k -> k + 1)\) - 1)\)/\((ratn - 
                    1)\)]; 
u = 
          Numerator[rat]; 
v = 
          Denominator[rat]; 
If[
          Not[PolynomialQ[u, k] && PolynomialQ[v, k]], 
          Return["Eingabe ist kein hypergeometrischer Term"]]; 
M = 
          DispersionsMenge[u /. k -> k - 1, v, 
            k]; 
dis = Max[M]; 
h = 
          PolynomialGCD[Product[u /. k -> k - 1 - j, {j, 0, dis}], 
            Product[v /. k -> k + j, {j, 0, 
                dis}]]; 
If[dis < 1, 
          A = Together[
              u/\((h /. k -> 
                      k + 1)\)]; 
B = \(-Together[v/h]\); \ 
          CC = v, gcd = 
            PolynomialGCD[h*u, \(-\((h /. k -> k + 1)\)\)*v, 
              h*\((h /. k -> k + 1)\)*v]; 
A = 
            Together[h*u/gcd]; \ 
          B = Together[\(-\((h /. k -> k + 1)\)\)*v/gcd]; \ 
          CC = Together[
              h*\((h /. k -> k + 1)\)*v/gcd]]; 
g = 
          REtoPol[A, B, CC, k]; 
If[
          g === "keine Polynomlösung", 
          Return["WZ-Methode scheitert"]]; 
SimplifyCombinatorial[g/h*\((ratn - 1)\)]]\)

WZCertificate[((Binomial[n, k] (x^k) (y^(n - k)))/(((x + y))^n)), 
      k, n]

WZCertificate[((k Binomial[n, k])/(n (2^(n - 1)))), k, 
      n]

\(WZCertificate[Binomial[n, k]\^2\/Binomial[2  n, n], k, n]\)

WZCertificate[
      HyperTerm[{-n, a}, {b}, 1, k]/(Pochhammer[b - a, n]/Pochhammer[b, n]), k, n]