a = (((-1))^k) Binomial[n, k]

rat = FunctionExpand[((a /. k -> k + 1)/a)]

u = Numerator[rat];  v = Denominator[rat];

h = 1;

RE = 
      h*u*g[k + 1] - (h /. k -> k + 1)*v*g[k] == 
        h*(h /. k -> k + 1)*v

RE2 = RE /. {g[k_] -> A + B k} // ExpandAll

gleichungen = 
      CoefficientList[RE2[[1]] - RE2[[2]], k] == 0

lösung = Solve[gleichungen, {A, B}]

g = A + B k /. lösung[[1]]

s = (g/h) a

(s /. k -> n + 1) - (s /. k -> 0)

Needs["SpecialFunctions`"]

Gosper[a, k]

Clear[g]

a = k k!

rat = FunctionExpand[((a /. k -> k + 1)/a)]

u = Numerator[rat];  v = Denominator[rat];

h = PolynomialGCD[(u /. k -> k - 1), v]

RE = 
      h*u*g[k + 1] - (h /. k -> k + 1)*v*g[k] == 
        h*(h /. k -> k + 1)*v

RE2 = RE /. {g[k_] -> A} // ExpandAll

gleichungen = 
      CoefficientList[RE2[[1]] - RE2[[2]], k] == 0

Solve[gleichungen, {A}]

Gosper[a, k]

a = k!

rat = FunctionExpand[((a /. k -> k + 1)/a)]

u = Numerator[rat];  v = Denominator[rat];

h = 1;

RE = 
      h*u*g[k + 1] - (h /. k -> k + 1)*v*g[k] == 
        h*(h /. k -> k + 1)*v

Gosper[a, k]