With the help of @JimBaldwin I found this solution

    (*Our pseudo-generating function*)
g[x_, m_, c_, bigD_, t_, d_] := ((c/bigD)/x + (bigD - t - c)/bigD + (t - d) x/bigD + (d/bigD) x^2)^m

    (*Probability mass function of k*)
f[k_, m_, c_, bigD_, t_, d_] := If[k == 0,Sum[Coefficient[g[x, m, c, bigD, t, d], x, i], {i, -m, 0}], Coefficient[g[x, m, c, bigD, t, d], x, k]]

    (*We make a probability distribution*)
dist[m_, bigD_: 6, t_: 2, d_: 0, c_: 0] := ProbabilityDistribution[f[k, m, c, bigD, t, d], {k, 0, 2 m, 1}, Assumptions -> {bigD \[Element] Integers, 0 <= d < bigD, 0 <= c < bigD, 1 <= t < bigD, m \[Element] Integers && m > 0}];

    (* Testing some statistics*)
{Mean[#], StandardDeviation[#], Table[Evaluate@CDF[#, k], {k, 0, 5}]} &@dist[5]

    (* Some plots *)
DiscretePlot[Evaluate@#1[dist[10], #2], {k, 0, 10}, ExtentSize -> 3/4] & @@@ {{PDF, k}, {SurvivalFunction, k - 1}}

Because this probability function is modeling a dice game it could be a difference depending in how we can interpret the parameter $d$ what could imply that $K\in\{0,\ldots,M\}$ strictly instead that $K\in\{0,\ldots,2M\}$ when $d>0$.

With the help of @JimBaldwin I found this solution

    (*Our pseudo-generating function*)
g[x_, m_, c_, bigD_, t_, d_] := 
 ((c/bigD)/x + (bigD - t - c)/bigD + (t - d) x/bigD + (d/bigD) x^2)^m

    (*Probability mass function of k*)
f[k_, m_, c_, bigD_, t_, d_] := 
 If[k == 0,Sum[Coefficient[g[x, m, c, bigD, t, d], x, i], {i, -m, 0}],
  Coefficient[g[x, m, c, bigD, t, d], x, k]]

    (*We make a probability distribution*)
dist[m_, bigD_: 6, t_: 2, d_: 0, c_: 0] := 
 ProbabilityDistribution[f[k, m, c, bigD, t, d],{k, 0, 2 m, 1}, 
  Assumptions -> {bigD \[Element] Integers, 0 <= d < bigD, 0 <= c < bigD, 1 <= t < bigD, 
  m \[Element] Integers && m > 0}];

    (* Testing some statistics*)
{Mean[#], StandardDeviation[#], Table[Evaluate@CDF[#, k], {k, 0, 5}]} &@dist[5]

    (* Some plots *)
DiscretePlot[Evaluate@#1[dist[10], #2], {k, 0, 10}, ExtentSize -> 3/4] & @@@ 
 {{PDF, k}, {SurvivalFunction, k - 1}}

Because this probability function is modeling a dice game it could be a difference depending in how we can interpret the parameter $d$ what could imply that $K\in\{0,\ldots,M\}$ strictly instead that $K\in\{0,\ldots,2M\}$ when $d>0$.